Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Power Means Calculus
Product Calculus,
Harmonic Mean Calculus, and
Quadratic Mean Calculus
H. Vic Dannon
vick@adnc.com
March, 2008
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Abstract
Each Power Mean of order r ≠ 0 ,
(
a1r +a2r +...anr
n
1
r
)
is associated with a Power Mean Derivative of order r , D (r ) .
We describe the
Arithmetic Mean Calculus obtained if r = 1 ,
Geometric Mean Calculus obtained if r → 0 ,
Harmonic Mean Calculus obtained if r = −1 ,
Quadratic Mean Calculus obtained if r = 2
Keywords
Calculus, Power Mean, Derivative, Integral, Product
Calculus. Gamma Function,
Mathematics Subject Classification
26A42, 26A24, 46G05,
2
26A06, 26B12, 33B15,
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Contents
Introduction..…………………………………………………………………………9
1. Arithmetic Mean Calculus........................................................................11
1.1 The Arithmetic Mean of f (x ) over [a, b ] …………………………………......11
1.2 Mean Value Theorem for the Arithmetic Mean…………………………......12
1.3 The Arithmetic Mean of f (x ) over [x , x + dx ] ………………….................12
1.4 Arithmetic Mean Derivative……………………………………………………13
1.5 The Arithmetic Mean Derivative is the Fermat-Newton-Leibnitz
Derivative……….…………………………………………………………………13
1.6 The Arithmetic Mean Derivative is an Additive Operator…………….......14
1.7 The Arithmetic Mean Derivative is not a Multiplicative Operator………14
2. The Product Integral…………………………………………………………..15
2.1 Growth problems…………………………………………………………………15
2.2 The Product Integral of e r (t ) over [a, b ] ……………………………..……….15
2.3 The Product Integral of f (x ) over [a, b ] ………………………….…………..16
2.4 Intermediate Value Theorem for the Product Integral…………………….17
2.5 The Product Integral is a Multiplicative Operator…………………………18
3. Geometric Mean and Geometric Mean Derivative.............................19
3.1 The Power Mean with r → 0 is the Geometric Mean……………..……...19
3.2 The Geometric Mean of f (x ) over [a, b ] ……………………………..……...20
3.3 Mean Value Theorem for the Geometric Mean……………………………..20
3.4 The Geometric Mean of f (x ) over [x , x + dx ] ……………………………...21
3.5 Geometric Mean Derivative……………………………………………………21
3
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
3.6 D (0)G (x ) = e D log G (x ) …………………………………………………………..22
(0)
3.7 D G (x ) = e
DG (x )
G (x )
……………………………………………………………..22
3.8 The Geometric Mean Derivative is non-additive operator………………..22
3.9 The Geometric Mean Derivative is multiplicative operator………………23
3.10 Geometric Mean Derivative Rules……………………………………..…….23
4. Geometric Mean Calculus……………………………………………….…...24
4.1 Fundamental Theorem of the Product Calculus………………………..…...24
4.2 Table of Geometric Mean Derivative, and Product Integrals……………..24
5. Product Differential Equations……………………………………..………26
5.1 Product Differential Equations……………………………………….………..26
5.2 Product Calculus Solution of
5.3
dy
dx
dy
dx
= P (x )y ……………………………...…..27
= P (x )y + Q(x ) may not be solved by Product Calculus.…………....27
5.4 y '' = P (x )y '+ Q(x )y may not be solved by Product Calculus.................29
6. Product Calculus of sin x …………………………….…………………….....30
x
6.1 Euler’s Product Representation for sin z …..…………………………………30
z
6.2 Conversion to Trigonometric Series…………………………………..……….30
6.3 Geometric Mean Derivative of sin x ..……………………………...…………..31
x
6.4 Second Geometric Mean Derivative of sin x ..……………………….………..32
x
6.5 Product Integration of sin x ..……………………………………………………32
x
6.6 Euler’s 2nd Product Representation for sin z …..……………………………..34
z
4
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
6.7 Geometric Mean Derivative of Euler’s 2nd product for sin x .......................35
x
7. Product Calculus of sin x ………………………..……………………………36
7.1 Euler’s Product Representation for sin z …..……………………...………...36
7.2 Geometric Mean Derivative of sin x ………………………………...………..39
7.3 The Wallis Product for π ……………………………………………………….39
8. Product Calculus of cos x …………………………………………...………..41
8.1 Euler’s Product Representation for cos z ………………………………..…..41
8.2 Geometric Mean Derivative of cos x ……………………………..…………..41
9. Product Calculus of tan x …………………………………………..………..43
9.1 Product Representation for tan z ……………...……………………..….……43
9.2 Geometric Mean Derivative of tan x ……………………………..…….…….43
10. Product Calculus of sinh x ………………………..……………..…………45
10.1 Product Representation of sinh z ………………………………...………….45
10.2 Geometric Mean Derivative of sinh x ……………………………..………..45
11. Product Calculus of cosh x …………………………………...……….……46
11.1 Product Representation of cosh z ……………………………………………46
11.2 Geometric Mean Derivative of cosh x ……………………………………....46
12. Product Calculus of tanh x ………………………………….....……..……47
12.1 Product Representation for tanh x ……………………………..…………..47
12.2 Geometric Mean Derivative of tanh x …………………………..………….47
13. Product Calculus of e x ………………………..………………………..……49
13.1 Product representation of e x ………………………………………………….49
13.2 Geometric Mean Derivative of e x ……………………………………………49
5
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
x
13.3 Geometric Mean Derivative of ee ……………………………………..…...50
k
13.4 Geometric Mean Derivative of e x …………………………………………..50
14. Geometric Mean Derivative by Exponentiation…………………..…52
14.1 D (0) sin x = e cot x ………………………………………………………………52
14.2 D (0)x x = ex ………………………………………………………….………...53
15. Product Calculus of Γ(x ) ……………………………………………………55
15.1 Euler’s Product Representation for Γ(x ) …………………………………...56
15.2 Geometric Mean Derivative of Γ(x ) …………………………………………58
15.3 Γ(1) = 1 ………………………………………………………………………….60
15.4 Γ(z + 1) = z Γ(z ) ………………………………………………………………60
15.5 Product Reflection Formula for Γ(z ) ……………………………………….61
15.6 Γ(z )Γ(1 − z ) =
15.7
2
( Γ(21) )
π
sin πz
…………………………..……………………………...61
= π ……………………………………………………………………62
2
15.8 ⎡⎢ Γ 1 ⎤⎥ = 2 ⋅ 2 2 4 4 6 6 8 8 10 10 12 12 .... ……………………….……..……62
1 3 3 5 5 7 7 9 9 11 11 13
⎣ 2 ⎦
( )
16. Products of Γ(z ) ………………………………………………………………64
16.1
Γ(1+z1 )
Γ(1+w1 )Γ(1+w2 )
, where z1 = w1 + w2 .......................................................65
16.2
Γ(1)
Γ(1+ix )Γ(1−ix )
16.3
Γ(1+ z1 )Γ(1+z 2 )
,
Γ(1+w1 )Γ(1+w2 )Γ(1+w 3 )
16.4
Γ(1+z1 )Γ(1+ z 2 )...Γ(1+ z k )
..............................................................................66
Γ(1+w1 )Γ(1+w2 )....Γ(1+wl )
= sinh x ............................................................................65
where z1 + z 2 = w1 + w2 + w 3 ……………….66
6
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
17. Product Calculus of J ν (x ) ………………………………………………..…67
17.1 Product Formula for J ν (z ) ……………………………………………………67
17.2 Geometric Mean Derivative of J ν (z ) ……………………………………….67
18. Product Calculus of Trigonometric Series………………….…………69
18.1 Product Integral of a Trigonometric Series…………………………………69
19. Infinite Functional Products………………………………………………70
19.1 Geometric Mean Derivative of an Euler Infinite Product……………….70
20. Path Product Integral………………………………………………………..71
20.1 Path Product Integral in the Plane…………………………………………..71
20.2 Green’s Theorem for the Path Product Integral………..………………….71
20.3 Path Product Integral in E 3 ………………………………………………….72
20.4 Stokes Theorem for the Path Product Integral…………………………….72
21. Iterative Product Integral…………………………………………………..73
21.1 Iterative Product Integral of f (x , t ) ………………………………………..73
21.2 Iterative Product Integral of e r (x ,t )dx .......................................................73
22. Harmonic Mean Integral…………………………………………………….74
22.1 Harmonic Mean Integral………………………………………………………74
23. Harmonic Mean and Harmonic Mean Derivative……………………75
23.1 The Harmonic Mean of f (x ) over
[a,b ]...................................................75
23.2 Mean Value Theorem for the Harmonic Mean…………………………….76
23.3 The Harmonic Mean of f (x ) over [x , x + dx ] .........................................76
23.4 Harmonic Mean Derivative……………………………………………………77
7
Gauge Institute Journal, Volume 4, No 4, November 2008
23.5 D (−1)H (x ) =
H. Vic Dannon
1
….………………………………………………………..77
Dx H (x )
24. Harmonic Mean Calculus……………………………………………………78
24.1 The Fundamental Theorem of the Harmonic Mean Calculus…………...78
24.2 Table of Harmonic Mean Derivatives and Integrals………………………78
25. Quadratic Mean Integral.…………………………………………………...80
25.1 Quadratic Mean Integral……………………………………………………...80
25.2 Cauchy-Schwartz Inequality for Quadratic Mean Integrals……….……81
25.3 Holder Inequality for Quadratic Mean Integrals …………………………81
26. Quadratic Mean and Quadratic Mean Derivative…………………...83
26.1 Quadratic Mean of f (x ) over
[a,b ]...........................................................83
26.2 Mean Value Theorem for the Quadratic Mean…………………………….84
26.3 The Quadratic Mean of f (x ) over [x , x + dx ] ........................................84
26.4 Quadratic Mean Derivative…………………………………………………..84
1/2
26.5 D (2)Q(x ) = ( DxQ(x ) )
……………………………………………………..85
27. Quadratic Mean Calculus…………………………………………………..86
27.1 The Fundamental Theorem of the Quadratic Mean Calculus…………..86
27.2 Table of Quadratic Mean Derivatives and Integrals……………………..86
References……………………………………………………………………………88
8
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Introduction
We describe a generalized calculus that was suggested by Michael
Spivey’s [Spiv] observation of the relation between the Geometric
Mean of a function over an interval, and its product integral.
We will see that each Power Mean of order r ≠ 0 ,
(
a1r +a2r +...anr
n
)
1
r
is associated with a Power Mean Derivative of order r ,
D (r ) .
The Fermat/Newton/Leibnitz Derivative
D (1) = D ≡
d
dx
is associated with the Arithmetic Mean
a1 +a2 +...an
n
,
which is Power Mean of order r = 1 .
The Geometric Mean Derivative
D (0)
is associated with the Geometric Mean
1/n
(a1 a2 ...an )
which is Power Mean of order r → 0 [Kaza].
9
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Product Integration is an operation inverse to the Geometric Mean
Derivative. Both are multiplicative operations, that apply
naturally to products, and in particular to Γ(z ) , the analytic
extension of the factorial function
The Harmonic Mean Derivative
D (−1)
is associated with the Harmonic Mean
n
1 1
1
+ +...
a1 a2
an
which is Power Mean of order r = −1 .
The Quadratic Mean Derivative
D (2)
is associated with the Power Mean of order r = 2 ,
(
The
inverse
operation,
a12 +a22 +...an2
n
the
)
1
2
.
Quadratic
Mean
Integration
transforms a function to its L2 norm squared.
We proceed with the definition of the Arithmetic Mean Derivative.
10
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
1
Arithmetic Mean Calculus
1.1
The Arithmetic Mean of f (x ) over [a,b ]
Given a function f (x ) that is Riemann integrable over the interval
[a,b ] , partition the interval into n sub-intervals, of equal length
b −a
,
n
choose in each subinterval a point
ci ,
Δx =
and consider the Arithmetic Mean of f (x ) ,
f (c1) + f (c2 ) + ...f (cn )
1
=
( f (c1) + f (c2 ) + ...f (cn ) ) Δx
n
b −a
As n → ∞ , the sequence of the Arithmetic Means converges to
x =b
1
b −a
∫
f (x )dx =
x =a
F (b )−F (a )
,
b −a
where
t =x
F (x ) =
∫
f (t )dt .
t =0
Therefore, the Arithmetic Mean of f (x ) over [a,b ] is defined by
x =b
1
b −a
∫
f (x )dx
x =a
11
Gauge Institute Journal, Volume 4, No 4, November 2008
1.2
H. Vic Dannon
Mean Value Theorem for the Arithmetic Mean
x =b
There is a point a < c < b , so that
1
b −a
∫
f (x )dx = f (c) .
x =a
t =x
Proof:
Since
∫
F (x ) =
f (t )dt
is
continuous
on
[a, b ] ,
and
t =0
differentiable in (a,b) , by Lagrange Intermediate Value Theorem
there is a point
a <c <b,
so that
F (b )−F (a )
b −a
= f (c) .
That is,
x =b
1
b −a
1.3
∫
f (x )dx = f (c) .
x =a
The Arithmetic Mean of f (x ) over [x , x + dx ]
The Arithmetic Mean of f (x ) over [x , x + dx ] is the Inverse
operation to Integration
Proof: By 1.2, there is
x < c < x + Δx ,
so that
t =x +Δx
1
Δx
∫
f (t )dt = f (c)
t =x
Letting Δx → 0 , the Arithmetic Mean of f (x ) at x , equals f (x ) .
12
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
t =x +Δx
lim Δ1x
Δx → 0
∫
f (t )dt = f (x ) .
t =x
Thus, the operation of finding the Arithmetic Mean of f (x ) x , is
inverse to integration.
This leads to the definition of the Arithmetic Mean Derivative.
t =x
1.4
Arithmetic Mean Derivative of F (x ) =
∫
f (t )dt at x
t =0
The Arithmetic Mean Derivative of
t =x
∫
F (x ) =
f (t )dt
t =0
at x is defined as the Arithmetic Mean of f (x ) over [x , x + dx ]
t = x +Δx
∫
D (1)F (x ) ≡ lim
1
Δx → 0 Δx
1.5
f (t )dt
t =x
The Arithmetic Mean Derivative is the FermatNewton-Leibnitz derivative
D (1)F (x ) =
dF (x )
dx
t = x +dx
Proof:
(1)
D F (x ) = Standard Part of
= Standard Part of
13
1
dx
∫
f (t )dt
t =x
F (x +dx )−F (x )
dx
Gauge Institute Journal, Volume 4, No 4, November 2008
=
1.6
dF (x )
dx
H. Vic Dannon
= DF (x ) .
The Arithmetic Mean Derivative
is an
Additive
Operator
D ( F1(x ) + F2 (x ) ) = DF1(x ) + DF2(x )
Thus, the Arithmetic Mean Derivative applies effectively to
infinite series.
1.7 The Arithmetic Mean Derivative is not a multiplicative
operator
D ( F1(x )F2(x ) ) = ( DF1(x ) ) F2 (x ) + F1(x )( DF2 (x ) )
Thus, Arithmetic Mean Derivative does not apply easily to infinite
products.
14
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
2
The Product Integral
2.1 Growth Problems
The Arithmetic Mean Derivative is unsuitable when we are
interested in the quotient
Present Value
Invested Value
Similarly, attenuation or amplification is measured by
Out-Put Signal
.
In-Put Signal
The need for a multiplicative derivative operator motivated the
creation of the product integration.
2.2
The Product Integral of e r (t ) over the interval [a, b ]
An amount A compounded continuously at rate r (t ) over time dt
becomes
Ae r (t )dt .
Over n equal sub-intervals of the time interval [a,b ] ,
Δt =
b −a
n
,
we obtain the sequence of finite products
Ae r (t1 )Δte r (t2 )Δt ...e r (tn )Δt = Ae[r (t1 )+r (t2 )+...+r (tn )]Δt .
15
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
As n → ∞ , the sequence converges to
t =b
∫
r (t )dt
Ae t =a
The amplification factor
t =b
∫
r (t )dt
e t =a
is called
the product integral of e r (t ) over the interval [a,b ]
and is denoted
t =b
∏ er (t )dt .
t =a
Thus, the Product Integral of e r (t ) over the interval [a,b ] is
t =b
t =b
∏e
r (t )dt
≡e
∫
r (t )dt
t =a
t =a
2.3
The Product Integral of f (x ) over the interval [a, b ]
Given a Riemann integrable, positive f (x ) on [a, b ] , partition the
interval into n sub-intervals, of equal length
Δx =
b −a
,
n
choose in each subinterval a point
ci ,
and consider the finite products,
16
Gauge Institute Journal, Volume 4, No 4, November 2008
Δx
f (c1 )Δx f (c2 )Δx ...f (cn )Δx = e ln[ f (c1 )
H. Vic Dannon
f (c2 )Δx ...f (cn )Δx ]
= e[ln f (c1 )+ ln f (c2 )+...+ ln f (cn )]Δx .
As n → ∞ , the sequence of products converges to
x =b
e
∫ ( ln f (x ) )dx
x =a
> 0.
We call this limit
the product integral of f (x ) over the interval [a, b ] ,
and denote it by
x =b
∏ f (x )dx .
x =a
Thus, the Product Integral of f (x ) over the interval [a,b ] is
x =b
x =b
∏ f (x )
dx
≡e
∫ ( ln f (x ) )dx
x =a
x =a
2.4 Intermediate Value Theorem for the Product Integral
x =b
There is a point a < c < b , so that
∏ f (x )dx
= f (c)(b −a )
x =a
t =x
Proof:
Since ϕ(x ) =
∫ ( ln f (t ))dt
is continuous on [a,b ] , and
t =0
differentiable in (a, b) , by Lagrange Intermediate Value Theorem
there is a point
17
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
a <c <b,
so that
x =b
∫ ( ln f (x ))dx = ϕ(b) − ϕ(a ) = ( ln f (c) )(b − a ) .
x =a
Hence,
x =b
e
∫ ( ln f (x ))dx
x =a
= e(
ln f (c ) )(b −a )
= f (c)(b −a ) .
2.5 The Product Integral is a multiplicative
operator
x =b
If a < c < b ,
∏ f (x )
dx
x =a
x =b
⎛ x =c
⎞⎛
⎞⎟
⎟
⎜
⎜
dx
dx
⎟
= ⎜⎜ ∏ f (x ) ⎟ ⎜⎜ ∏ f (x ) ⎟⎟
⎟⎟ ⎜
⎟
⎜⎝ x =a
⎠⎝
⎠⎟
x =c
Proof: If a < c < b ,
x =c
x =b
∏ f (x )
dx
=e
x =b
∫ ( ln f (x ))dx + ∫ ( ln f (x ) )dx
x =a
x =c
x =a
⎛ x =c
⎞⎛ x =b
⎞
⎜
⎜
dx ⎟
dx ⎟
⎟
⎜
⎜
= ⎜ ∏ f (x ) ⎟ ⎜ ∏ f (x ) ⎟⎟ .
⎟
⎜⎝ x =a
⎟⎟ ⎜ x =c
⎠⎝
⎠⎟
The inverse operation to product integration is the Geometric
Mean Derivative.
18
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
3
Geometric
Mean
and
Geometric
Mean Derivative
3.1 The Power Mean with r → 0 is the Geometric
Mean
(
a1r
+a2r +...anr
n
1
r
)
1
n
→ ( a1a2 ...a 3 )
r →0
Proof: Let r → 0 in
(
a1r
+a2r +...anr
n
Then, the exponent
1
r
1
r
)
1 ⎡ log a r +a r +...a r − log n ⎤
⎥⎦ .
= e r ⎢⎣ ( 1 2 n )
⎡ log(a r + a r + ...a r ) − log n ⎤ is of the form
n
1
2
⎣⎢
⎦⎥
and by L’Hospital, its limit is
lim
Dr { log(a1r + a2r + ...anr ) − log n }
Dr r
r →0
1
= lim
r
r
r
r → 0 a1 +a2 +...an
=
1
n
(a1r ln a1 + a2r ln a2 + ...anr ln an )
( ln a1 + ln a2 + ... + ln an )
1
n
= ln ( a1a2...an ) .
Therefore,
(
a1r
+a2r +...anr
n
1
r
)
1
⎯⎯
⎯→ e
r →0
ln(a1a2 ...an )n
19
1
= ( a1a2 ...an )n .
0,
0
Gauge Institute Journal, Volume 4, No 4, November 2008
3.2
H. Vic Dannon
f (x ) over [a,b ]
Geometric Mean of
Given an integrable function f (x ) that is positive over [a,b ] ,
partition the interval, into n sub-intervals, of equal length
Δx =
b −a
,
n
choose in each subinterval a point
ci ,
and consider the Geometric Mean of f (x )
1/n
( f (c1)f (c2 )...f (cn ) )
=
1
( ln f (c1 )+ ln f (c2 )+... ln f (cn ))Δx
e b −a
.
As n → ∞ , the sequence of Geometric Means converges to
e
1
b −a
x =b
∫ ( ln f (x ))dx
x =a
=
G (b )
,
G (a )
where
t =x
G (x ) = e
∫ ( ln f (t ))dt
t =a
Therefore,
x =b
e
1
( ln f (x ) )dx
b −a x∫=a
is defined as the Geometric Mean of f (x ) over [a,b ] .
3.3 Mean Value Theorem for the Geometric Mean
x =b
There is a point a < c < b , so that e
20
1
( ln f (x ) )dx
b −a x∫=a
= f (c)
Gauge Institute Journal, Volume 4, No 4, November 2008
Proof:
3.4
H. Vic Dannon
By 2.3, and 2.4.
The Geometric Mean of f (x ) over [x , x + dx ]
The Geometric Mean of f (x ) over [x , x + dx ] , is the Inverse
Operation to Product Integration
Proof: By 3.3, there is
x < c < x + Δx ,
so that
t = x +Δx
e
1
Δx
∫
ln f (t )dt
= f (c) .
t =x
Letting Δx → 0 , the Geometric Mean of f (x ) at x equals f (x ) .
t = x +Δx
lim e
1
Δx
∫
ln f (t )dt
= f (x )
t =x
Δx → 0
Thus, the operation of finding the Geometric Mean of f (x ) over
[x , x + dx ] , is inverse to product integration over [x , x + dx ] .
This leads to the definition of the Geometric Mean Derivative
3.5 Geometric Mean Derivative
The Geometric Mean Derivative of
t =x
G (x ) = e
∫ ( ln f (t ))dt
t =a
at x is defined as the Geometric Mean of f (x ) over [x , x + dx ]
21
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
t =x +Δx
(0)
D G (x ) ≡ lim e
1
Δx
∫ ( ln f (t ) )dt
t =x
Δx → 0
D (0)G (x ) = e D log G (x )
3.6
t = x +Δx
Proof:
(0)
D G (x ) = lim e
∫ ( ln f (t ))dt
1
Δx
t =x
Δx → 0
t = x +Δx
1
Δx → 0 Δx
lim
=e
∫ ( ln f (t ))dt
t =x
t = x +dx
= Standard part of e
= Standard Part of
∫ ( ln f (t ))dt
1
dx
(
t =x
1
G (x +dx ) dx
G (x )
)
1
⎡ log G (x +dx )− log G (x ) ⎤
⎦
= Standard Part of e dx ⎣
= e D log G (x ) .
3.7
D (0)G (x ) = e
DG (x )
G (x )
3.8 The Geometric Mean Derivative is non- additive operator
D (G1(x )+G2 (x ) )
Proof: D (0)(G1 + G2 )(x ) = e
G1(x )+G2 (x )
22
.
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
3.9 The Geometric Mean Derivative is a multiplicative operator
Proof: D (0) ⎣⎡G1(x )G2(x ) ⎦⎤ = e D log[G1(x )G2 (x )]
(
=e
=e
DG1 ( x ) DG2 ( x )
+
G1 ( x )
G2 ( x )
)
DG1 ( x ) DG2 ( x )
G1 ( x )
G2 ( x )
e
= ( D (0)G1(x ) )( D (0)G2 (x ) ) .
3.10 Geometric Mean Derivative Rules
2
( D(0) ) G(x ) ≡ D(0)eD lnG(x ) = eD
(
D (0)
n
)
2 lnG (x )
n
G (x ) = e D lnG (x )
D (0) ( f (x )g (x ) ) = f (x )Dg (x )e g (x )D ln f (x )
D (0) f (g(x )) = e
23
df dg
dg dx
f (g (x ))
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
4
Geometric Mean Calculus
4.1 The Fundamental Theorem of the Product Calculus
D
(0)
t =x
∏ f (t )dt
= f (x )
t =a
t =x
Proof: Dx
(0)
t =x
∏ f (t )
t =a
dt
(0)
= Dx e
∫ ( ln f (t ))dt
t =a
1
=e
t =x
∫ ( ln f (t ) )dt
e t =a
Dx
t =x
∫ ( ln f (t ) )dt
t
=
e a
t =x
t =x
∫ ( ln f (t ) )dt
t
a
=
e
Dx
ln f (t )
t =a
t =x
∫ ( ln f (t ) )dt
et =a
∫(
=e
)dt
t =x
=e
Dx
∫ ( ln f (t ) )dt
t =a
= f (x ) .
4.2 Table of Geometric Mean Derivatives and integrals
24
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
We list some geometric Mean Derivatives, and Product Integrals.
Some of these are given in [Spiv].
f (x )
∏
x
f (x )
D (0) f (x )
1
2
a
e
x
x2
xa
x −1
1
1
1
1
1
2x
ax
ex
e1/x
e2/x
ea /x
e−1/x
ex (ln x −1) = e−x x x
e2x (ln x −1) = e−2x x 2x
eax (ln x −1) = e−ax x ax
e−x (ln x −1) = ex x x
e1/x log x
e ∫ ln( ln x )dx
e2
ea
ex
ln x
e2x
eax
xx
n
ex
enx
sin x
e cot x
ex
n −1
I (0)f (x ) =
eax
2
2 /2
x2 x2
2
−
2
ex (ln x −1/2)/2 = e 4 x
n +1
ex /(n +1)
ln(sin x )dx
e∫
ln(cos x )dx
e∫
cos x e− tan x
e∫
tan x e2/sin2x
x
x
ee
ee
e sin x e cos x
e cos x e− sin x
ln(tan x )dx
ee
x
e− cos x
e sin x
25
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
5
Product Differential Equations
5.1 Product Differential Equations
A Product Differential Equation involves powers of the Geometric
Mean Derivative operator
D (0) ,
and no sums, only products.
An ordinary differential equations involves sums of powers of the
Arithmetic Mean Derivative operator D . Such equation is not
suitable to the application of D (0) , and does not convert easily into
a product differential equation.
[Doll] attempts to write the solutions to ordinary differential
equations in terms of product integral, but being unaware of the
Geometric Mean Derivative, it fails to produce one product
differential equation.
[Doll] demonstrates that products integrals are not natural
solutions for ordinary differential equations.
The attempt made in [Doll] to interpret Summation Calculus in
terms of the Product Integral alone, being oblivious to the product
Calculus derivative, does not lead to better understanding of
differential equations, or to any new results.
26
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Only the basic equation
dy
= P (x )y
dx
may be converted to a product differential equation, and be solved
as such.
dy
= P (x )y .
dx
5.2 Product Calculus Solution of
Dividing both sides by y(x ) ,
y'
= P (x ) .
y
e
y'
y
= e P (x )
D (0)y = e P (x )
t =x
t =x
y(x ) =
∏e
P (t )dt
=e
∫
P (t )dt
t =0
.
t =0
5.3
dy
= P (x )y + Q(x ) may not be solved by Product Calculus
dx
Proof: We don’t know of a Product Calculus method to solve the
equation
dy
= P (x )y + Q(x ) .
dx
27
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
t =x
In Arithmetic Mean Calculus, we multiply both sides by e
∫
t =0
P (t )dt
.
Then,
t =x
y 'e
∫
t =x
P (t )dt
+ yP (x )e
t =0
∫
t =x
P (t )dt
= Q(x )e
t =0
t =x
∫
P (t )dt
t =0
t =x
∫ P (t )dt
∫ P (t )dt
d
t
=
t
0
(ye
) = Q(x )e = 0
dx
t =x
ye
∫
t =0
t =u
u =x
P (t )dt
=
∫
Q(u )e
∫
P (t )dt
t =0
u =0
t =u
u =x
∫
y=
Q(u )e
∫
P (t )dt
t =0
u =0
t =x
e
∫
P (t )dt
t =0
Writing this as
u =x
∫
y=
t =x
Q(u )∏ e P (t )dt
t =0
u =0
t =x
∏e P (t )dt
t =0
demonstrates why the equation cannot be converted into a product
differential equation, and cannot be solved as such: In product
Calculus we need to have pure products. No summations.
28
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
5.4 y '' = P (x )y '+ Q(x )y may not be solved by Product Calculus
Proof: We may write
y '' = P (x )y '+ Q(x )y
as a first order system
y' = z
z ' = P (x )z + Q(x )y
Thus, in matrix form,
1 ⎞⎛
d ⎛⎜ y ⎞⎟ ⎛⎜ 0
⎟⎟⎜⎜ y ⎞⎟⎟ .
⎜⎜ ⎟⎟ = ⎜⎜
⎟⎟⎜ z ⎟⎟
⎜ ⎠
dx ⎝⎜ z ⎠⎟ ⎝⎜Q(x ) P (x ) ⎠⎝
But we need the methods of summation Calculus, to obtain two
independent solutions y1(x ) , and y2 (x ) that span the
space for the equation.
29
solution
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
6
Product Calculus of sinx
x
sin z
z
6.1 Euler’s Product Representation for
For any complex number z ,
sin z
= cos z2 cos z4 cos z8 ...
z
Proof: sin z = 2 cos z2 sin z2
= 2 cos z2 2 cos z4 sin z4
= 2 cos z2 2 cos z4 2 cos z8 sin z8
= 2 cos z2 2 cos z4 2 cos z8 ...2 cos
(
n
= z 2z sin
z
2n
z
2n
sin
z
2n
) cos z2 cos z4 cos z8 ...cos 2z .
n
Therefore, for any complex number z ≠ 0 ,
⎛ sin z ⎞⎟ 2zn
⎜⎜
= cos z2 cos z4 cos z8 ...cos zn
⎟
2
⎜⎝ z ⎠⎟ sin z
2n
Letting n → ∞ ,
sin z
= cos z2 cos z4 cos z8 ...
z
This holds also for z → 0 . Hence, it holds for any complex number
z.
6.2
Conversion to Trigonometric Series
30
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Products of Cosines can be converted into summations, and the
infinite product may be converted into a Trigonometric Series.
For instance,
cos α cos β cos γ = ( 21 cos(α + β ) + 21 cos(α − β ) ) cos γ
= 21 cos(α + β )cos γ + 21 cos(α − β )cos γ
= 14 cos(α + β + γ ) + 14 cos(α + β − γ )
+ 14 cos(α − β + γ ) + 14 cos(α − β − γ )
6.3
Geometric Mean Derivative of
sin x
x
cos x 1
− = − 21 tan x2 − 12 tan x2 − 13 tan x3 − ...
2
2
2
2
sin x x
Proof: Geometric Mean Differentiating both sides of 6.1,
D (0)
e
e
sin x
= ( D (0) cos x2 )( D (0) cos x4 )( D (0) cos x8 )...
x
⎛ sin x ⎞⎟
⎜⎜ D
⎟
⎜⎜
x ⎟⎟
⎜⎜ sin x ⎟⎟⎟
⎟
⎜⎜
⎝ x ⎠⎟
cos x − 1
sin x x
=e
⎛ D cos x ⎞⎟
⎜⎜
2⎟
⎟⎟
⎜⎜
x ⎟
cos
⎜⎝
⎠
2
=e
e
⎛ D cos x ⎞⎟
⎜⎜
22 ⎟
⎟⎟
⎜⎜
⎜⎜ cos x ⎟⎟⎟
⎝
22 ⎠
e
⎛ D cos x ⎞⎟
⎜⎜
23 ⎟
⎟⎟
⎜⎜
⎜⎜ cos x ⎟⎟⎟
⎝
23 ⎠
...
1
x
1
x
−21 tan x2 −22 tan 22 −23 tan 23
e
e
That is,
cos x 1
− = − 21 tan x2 − 12 tan x2 − 13 tan x3 − ...
2
2
2
2
sin x x
31
Gauge Institute Journal, Volume 4, No 4, November 2008
6.4
H. Vic Dannon
Second Geometric Mean Derivative of
−
Proof:
1
sin2 x
+
1
x2
=−
1
x
22 cos2
2
1
−
24 cos2
x
1
−
26 cos2
2
2
sin x
x
x
....
23
Second Geometric Mean Differentiation of 6.1 gives
⎛ cos x 1 ⎞⎟
⎜⎜
− ⎟
(0) ⎜⎝ sin x x ⎠⎟
D e
x
1
(0) ( − 2 tan 2 )
=D e
1
1
−
+ )
(
e sin x x = e
2
2
⎛
⎞⎟
⎜⎜
1
⎟⎟
−
⎜⎜ 2
2x ⎟
⎜⎝ 2 cos 2 ⎠⎟⎟
e
×D
(−
e
(0)
⎛
⎜⎜
1
⎜⎜ −
4
⎜ 2 cos2
⎝⎜
⎞⎟
⎟⎟
⎟
x ⎟
⎟
⎟
2⎠
2
e
1
22
) × ....
tan x
22
⎛
⎜⎜
1
⎜⎜ −
6
⎜ 2 cos2
⎝⎜
⎞⎟
⎟⎟
⎟
x ⎟
⎟
⎟
3⎠
2
....
That is,
−
1
sin2 x
+
1
x2
=−
1
x
22 cos2
2
1
−
24 cos2
x
2
2
−
1
26 cos2
x
....
23
The last series can be obtained by term by term
series-
differentiation of 6.3.
6.5
Product Integration of
∫ log
sin x
x
B
B
B6
B8
sin x
dx = − 2 (2x )3 + 4 (2x )5 −
(2x )7 +
(2x )9 − ....
x
4 ⋅ 3!
8 ⋅ 5!
12 ⋅ 7 !
16 ⋅ 9!
Where the B2, B4, B6,... are the Bernoulli Numbers.
32
Gauge Institute Journal, Volume 4, No 4, November 2008
Proof:
H. Vic Dannon
Product integrating 6.1,
log
e∫
sin x
dx
x
log cos x2dx ∫ log cos x4dx ∫ log cos x8dx
∫
=e
e
e
...
By [Grob, p.113, 8b], for x < π , up to a constant,
∫
log
sin x
dx =
x
=−
∫ log sin xdx − ∫ log xdx
B2
B
B6
B8
(2x )3 + 4 (2x )5 −
(2x )7 +
(2x )9 − .... ,
4 ⋅ 3!
8 ⋅ 5!
12 ⋅ 7 !
16 ⋅ 9!
where the B2, B4, B6,... are the Bernoulli Numbers.
By [Grob, p.113, 9b], for x < π , up to a constant,
∫ log cos x2 dx = 2∫ log cos x2 d x2
⎛ (22 − 1)B
⎞
(24 − 1)B4 5 (26 − 1)B6 7
⎜
3
2
(x ) +
(x ) −
(x ) + .... ⎟⎟⎟
= 2 ⎜⎜ −
4 ⋅ 3!
8 ⋅ 5!
12 ⋅ 7 !
⎝⎜
⎠⎟
∫ log cos x4 dx = 4 ∫ log cos x4 d x4
⎛ (22 − 1)B x
⎞
(24 − 1)B4 x 5 (26 − 1)B6 x 7
⎜
3
2
( ) +
( ) −
( ) + .... ⎟⎟⎟
= 4 ⎜⎜ −
⎜⎝
4 ⋅ 3!
2
8 ⋅ 5!
2
12 ⋅ 7 ! 2
⎠⎟
…………………………………………………………………………..
Comparing the coefficients of x 3, x 5, x 7 ,... on both sides, does not
yield any new result.
33
Gauge Institute Journal, Volume 4, No 4, November 2008
6.6
Euler’s 2nd Product for
H. Vic Dannon
sin z
z
For any complex number z
4 cos2 z3 − 1 4 cos2 3z2 − 1 4 cos2 3z3 − 1
sin z
=
×
×
× ...
z
3
3
3
Proof: Using the triple angle formula,
sin 3z = 3 sin z − 4 sin 3 z ,
[Zeid, p.57], we write
sin z = 3 sin z3 − 4 sin 3 z3
= sin z3 ( 3 − 4[1 − cos2 z3 ] )
= ( 4 cos2 z3 − 1 ) sin z3
(
= ( 4 cos2 z3 − 1 ) 4 cos2
z
32
(
)
− 1 sin
= ( 4 cos2 z3 − 1 ) × ... × 4 cos2
z
3n
z
32
)
− 1 sin
z
3n
4 cos2 zn − 1
4 cos2 z3 − 1
3n
3
x
sin n
=x
× ... ×
3
x
3
3
Therefore, for any complex number z ≠ 0 ,
z
4 cos2 zn − 1
4 cos2 z3 − 1
sin z 3n
3
=
× ... ×
z
z sin n
3
3
3
Letting n → ∞ ,
4 cos2 z3 − 1 4 cos2 3z2 − 1
sin z
=
×
× ...
z
3
3
34
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Since this holds also for z → 0 , it holds for any complex number
z including z = 0 .
6.7
Geometric Mean Derivative of Euler’s 2nd Product
Presentation for
sin x
x
4 sin 2x3
3
3
− = −
−
−
...
2 x
2
2 x
3
2 x
x
sin x
− 1) 3 (4 cos 2 − 1) 3 (4 cos 3 − 1)
3(4 cos
3
3
3
cos x
Proof:
4 sin 2x2
4 sin 2x
1
3
Geometric Mean Differentiating 6.6,
D
e
(0)
4 cos2 x2 − 1
4 cos2 x3 − 1
sin x
(0)
(0)
3
=D
×D
× ...
x
3
3
⎛ D sin x ⎞⎟
x ⎟
⎜⎜⎜
⎟
x ⎟
sin
⎜⎝
⎠⎟
x
=e
cos x − 1
(
)
e sin x x = e
⎛ 4D cos2 x ⎞⎟
⎜⎜
3 ⎟
⎟⎟
⎜⎜
2x
⎜⎝ 4 cos 3 −1 ⎠⎟
e
⎛
⎞
⎜⎜
4 sin 2 x ⎟⎟⎟
⎜⎜
⎟⎟
3
−
⎟
⎜⎜⎜
2 x −1) ⎟⎟⎟
3(4cos
⎜⎜⎝
⎠⎟
3
e
⎛ 4D cos2 x ⎞⎟
⎜⎜
32 ⎟
⎜⎜
⎟⎟⎟
2
x
⎜ 4 cos −1 ⎟⎟
2
⎝⎜
⎠
3
e
⎛
⎞⎟
⎜
4 sin 2 x
⎟⎟⎟
⎜⎜⎜
2
⎟⎟
3
⎜⎜ −
⎟
⎜⎜⎜ 32 (4 cos2 x −1) ⎟⎟⎟
⎟
⎜
⎝⎜
⎠⎟
32
⎛ 4D cos2 x ⎞⎟
⎜⎜
33 ⎟
⎜⎜
⎟⎟⎟
2
x
⎜ 4 cos −1 ⎟⎟
3
⎝⎜
⎠
e
3
...
⎛
⎞⎟
⎜
4 sin 2 x
⎟⎟⎟
⎜⎜⎜
3
⎟⎟
3
⎜⎜ −
⎟
⎜⎜⎜ 33 (4 cos2 x −1) ⎟⎟⎟
⎟
⎜
⎝⎜
⎠⎟
33
...
4 sin 2x2
4 sin 2x3
3
3
3
...
− =−
−
−
sin x
x
3(4 cos2 x − 1) 32 (4 cos2 x2 − 1) 3 3 (4 cos2 x3 − 1)
3
3
3
cos x
1
4 sin 2x
35
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
7
Product Calculus of
7.1
sinx
sinz
Euler’s Product Representation for
For any complex number z ,
⎛
z 2 ⎞⎛
z 2 ⎞⎛
z 2 ⎞⎟
⎟
⎟
⎜
⎜
⎜
⎟⎟⎜ 1 −
⎟⎟ ....
sin z = z ⎜⎜ 1 − 2 ⎟⎟ ⎜⎜ 1 −
⎟⎜
⎟⎜⎜
⎜⎝
(2π)2 ⎠⎝
(3π)2 ⎠⎟
π ⎠⎝
The product converges absolutely in any disk z < R .
Proof:
sin z = 2 sin z2 cos z2
= 2 ⋅ 2 sin z4 cos z4 sin ( z2 +
π
2
)
= 22 sin z4 sin ( z4 +
π
2
) sin ( z2 + π2 )
= 22 sin z4 sin ( z4 +
π
2
) 2 sin ( z4 + π4 ) cos ( z4 + π4 )
= 23 sin z4 sin ( z +42π ) sin ( z +4 π ) cos ( − z4 − π4 )
= 23 sin z4 sin ( z +42π ) sin ( z +4 π ) sin ( − z4 − π4 +
π
2
)
= 23 sin z4 sin ( z +42π ) sin ( z +4 π ) sin ( π −4 z )
= 27 sin z8 sin ( z +84 π ) sin ( z +4 π ) sin ( π −4 z ) ×
×sin ( z +82π ) sin ( 2π8−z ) sin ( z +8π 3 ) sin ( 3π8−z )
…………………………………………………………………………….
36
Gauge Institute Journal, Volume 4, No 4, November 2008
n
= 22
−1
sin
×sin
z
2n
n −1
sin z +2 n
π
H. Vic Dannon
sin z +nπ sin π −n z ×
2
2
2
( z +2 2π ) sin ( 2π2−z ) sin ( z +2 3π ) sin ( 3π2−z ) × ...
n
n
... × sin
n
z +(2n −1 −1)π
n
2
n
sin
(2n −1 −1)π −z
2n
Now,
(
sin z +nπ sin π −n z = 2 sin z n++π1 cos z n++π1
2
2
2
(
2
= 2 sin z n++π1 cos πn−+z1
2
(
= sin
= sin2
+ sin
π
2n
− sin2
z
2n
z
2n
π
2n
2
)( 2 sin 2π−z cos 2π−z )
n +1
n +1
)( 2 sin 2π−z cos z2+π )
n +1
n +1
)( sin 2π − sin 2z )
n
n
And,
z = sin2 2 π − sin2
sin z +n2π sin 2π −
n
n
2
2
2
z
2n
Therefore,
¤
n
sin z = 22
(
−1
sin
z
2n
cos
× sin2 2πn − sin2
2
z
2n
z
2n
( sin2 2π − sin2 2z ) ×
n
n
) × ... × ( sin2 (2
n −1
−1)π
2n
− sin2
z
2n
)
That is,
n
2
z
2n
sin
z
2n
n
sin z
= 22 −1 cos zn sin2 πn − sin2 zn ×
2
2
2
z
(
(
× sin2 2πn − sin2
2
)
z
2n
) × ... × ( sin2 (2
n −1
Letting z → 0 ,
37
−1)π
n
2
− sin2
z
2n
)
Gauge Institute Journal, Volume 4, No 4, November 2008
n
2n = 22
−1
sin2
π
2n
H. Vic Dannon
sin2 2πn × ... × sin2
2
(2n −1 −1)π
2n
Dividing ¤ by this last equation,
n
sin z = 2
⎛
sin2 zn
⎜⎜
2
1−
2 π
⎜⎜⎜
sin n
⎝
2
⎛
⎞⎛
sin2 zn ⎞⎟
sin2 zn ⎞⎟⎟
⎜⎜
⎟⎟ ⎜⎜
2 ⎟
2
⎟⎟⎟
n −1
⎟⎟⎟ ⎜⎜ 1 −
⎟⎟ × .. × ⎜⎜⎜ 1 −
2 2π ⎟
(2
1)
π
−
2
sin n ⎠⎟
⎟⎠⎟
⎟⎜
sin
⎜⎝
⎠⎝
2
2n
⎛
⎞⎟
sin2 zn
⎜
2
z
⎟⎟ cos ⎜⎜ 1 −
⎟⎟
2 π
2n ⎜
sin n
⎜⎝
⎠
2
⎛
⎞⎛
sin2 zn ⎞⎟
sin2 zn ⎞⎟⎟
⎜⎜
⎟⎟ ⎜⎜
2 ⎟
2
⎟⎟⎟
n −1
⎟⎟⎟ ⎜⎜ 1 −
⎟⎟ × .. × ⎜⎜⎜ 1 −
2 2π ⎟
(2
1)
π
−
2
sin n ⎠⎟
⎟⎜
sin
⎜⎝
⎠⎝
⎠⎟⎟
2
2n
sin zn
2
⎛ sin zn
⎜
2
= z ⎜⎜
z
⎝⎜⎜ 2n
cos zn
2
Letting n → ∞ , for any fixed natural number m we have
1−
sin2
z
2n
sin2 mnπ
2
⎛ sin zn
⎜
= 1 − ⎜⎜ z 2
⎜⎝⎜ n
2
⎞⎟2 z 2 ⎛ mnπ ⎞⎟2
z2
⎜⎜ 2 ⎟
⎟⎟
⎟ → 1−
⎟
mπ ⎟
2⎜
(m π)2
⎠⎟ (m π) ⎝⎜⎜ sin 2n ⎠⎟
Consequently, the infinite product
2
⎛
z 2 ⎞⎛
z 2 ⎞⎛
⎟⎟ ⎜⎜ 1 − z ⎞⎟⎟....
z ⎜⎜⎜ 1 − 2 ⎟⎟⎟⎜⎜⎜ 1 −
⎟⎟ ⎜⎜
⎟
⎟⎜
(2π)2 ⎠⎝
(3π)2 ⎠⎟
π ⎠⎝
⎝⎜
Converges to sin z .
The convergence is absolute in any disk z < R , because the
infinite series
z2
π2
+
z2
(2π)2
+
z2
(3π)2
+ ...
converges absolutely in any disk z < R .
38
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Indeed, in z < R ,
z
2
π2
+
z
2
(2π)2
+
2
z
(3π)2
+ ... = z
2
⎞⎟
1 ⎛⎜
1
1
+
+
+
1
...
⎟
⎜
⎠⎟
22 32
π 2 ⎜⎝
2
= z
1 π2
π2 6
1
< R2 .
6
7.2
Geometric Mean Derivative of
1
x
cot x = −
2x
π2 − x 2
−
2x
(2π)2 − x 2
−
sinx
2x
(3π)2 − x 2
− ...
⎛
⎛
x2 ⎞
x 2 ⎟⎞ (0) ⎛⎜
x 2 ⎞⎟
⎟⎟ D ⎜ 1 −
⎟ ....
Proof: D (0) sin x = D (0)xD (0) ⎜⎜ 1 − 2 ⎟⎟⎟ D (0) ⎜⎜ 1 −
2⎟
2⎟
⎜⎝
⎜
⎜
⎟
⎝
⎝
(2π) ⎠
(3π) ⎠⎟
π ⎠
e
cos x
sin x
1
x
=e e
−
2x
2
π −x 2
e
−
2x
(2 π )2 −x 2
e
−
2x
(3 π )2 −x 2
...
Thus,
1
x
cot x = −
7.3
2x
π2 − x 2
−
2x
(2π)2 − x 2
−
The Wallis Product for π
π
224466
=
....
2
133557
39
2x
(3π)2 − x 2
− ...
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Proof: Wallis product for π follows from the product formula for
sin π2 , [Bert, p. 424].
1 = sin π2
=
π ⎛⎜
1 ⎞⎛
1 ⎞⎛
1 ⎞⎟
⎟
⎟
⎜
⎜
−
−
−
1
1
1
⎟
⎟
⎟ ....
⎜
⎟⎜⎜
⎟⎜⎜
2 ⎜⎝
22 ⎠⎝
42 ⎠⎝
62 ⎠⎟
=
π ⎛⎜
1 ⎞⎛
1 ⎞⎛
1 ⎞⎛
⎜⎜ 1 − ⎟⎟⎟⎜⎜⎜ 1 + ⎟⎟⎟⎜⎜⎜ 1 − ⎟⎟⎟⎜⎜⎜ 1 +
2⎝
2 ⎠⎝
2 ⎠⎝
4 ⎠⎝
=
π133557
....
2224466
1 ⎞⎛
⎟⎟⎜⎜ 1 − 1 ⎞⎛
⎟⎟⎜⎜ 1 + 1 ⎞⎟⎟ ...
⎟⎜
⎟⎜
4 ⎠⎝
6 ⎠⎝
6 ⎠⎟
40
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
8
Product Calculus of
8.1
cos x
Euler’s Product Representation for
cos z ,
For any complex number z ,
2 ⎞⎛
2⎞
⎛
⎛ 2z ⎞⎟2 ⎞⎛
⎛
⎞
⎛
⎞
⎟
⎟
⎟
2
z
2
z
⎜⎜
⎜
⎜
cos z = ⎜ 1 − ⎜⎜ ⎟⎟ ⎟⎟⎟⎜⎜ 1 − ⎜⎜ ⎟⎟⎟ ⎟⎟⎟⎜⎜ 1 − ⎜⎜ ⎟⎟⎟ ⎟⎟⎟...
⎜⎝ π ⎠ ⎟⎜⎜
⎜⎜
⎝⎜ 3π ⎠ ⎠⎝
⎝⎜ 5π ⎠ ⎠⎟
⎟⎜⎜
⎝
⎠⎝
The convergence is absolute in any disk z < R .
Proof:
cos z =
=
sin 2z
2 sin z
2
2
⎛ ⎛ 2z ⎞2 ⎞⎛
⎞⎛ ⎛ 2z ⎞2 ⎞⎛
⎞⎛ ⎛ 2z ⎞2 ⎞⎟
⎛
⎛
2z ⎞ ⎟ ⎜
2z ⎞ ⎟ ⎜
⎟
⎟
⎜
⎜
⎜
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟
2z ⎜ 1 − ⎜ ⎟
⎜⎜ ⎝⎜ π ⎠⎟ ⎟⎟⎜⎜⎜ 1 − ⎝⎜⎜ 2π ⎠⎟ ⎟⎟ ⎜⎜⎜ 1 − ⎝⎜⎜ 3π ⎠⎟ ⎟⎟⎜⎜⎜ 1 − ⎝⎜⎜ 4 π ⎠⎟ ⎟⎟⎜⎜⎜ 1 − ⎝⎜⎜ 5π ⎠⎟ ⎟⎟ ...
1 ⎝
⎠⎝
⎠⎝
⎠⎝
⎠⎝
⎠
2
2
2
2
⎞⎛ ⎛ z ⎞2 ⎞⎛
⎞
2 ⎛
⎛ z ⎞⎟ ⎞⎟⎟⎛⎜ ⎛ z ⎞⎟ ⎞⎛
⎛
⎛
z ⎞ ⎟⎜
z ⎞ ⎟
⎟
⎟
⎜
⎜
⎜
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
z ⎜1 − ⎜
1
−
1
−
1
−
1
−
⎜⎜ ⎜⎝ π ⎟⎠ ⎟⎟⎜⎜ ⎝⎜ 2π ⎠⎟ ⎟⎟ ⎜⎜ ⎜⎝ 3π ⎠⎟ ⎟⎟ ⎜⎜ ⎝⎜ 4 π ⎠⎟ ⎟⎟⎜⎜ ⎝⎜ 5π ⎠⎟ ⎟⎟ ...
⎝
⎠⎝
⎠⎝
⎠⎝
⎠⎝
⎠
⎛
⎛ 2z ⎞⎟2 ⎞⎛
⎛ 2z ⎞⎟2 ⎞⎛
⎛ 2z ⎞⎟2 ⎞⎟⎟
⎟⎟⎜⎜
⎟⎟⎜⎜
⎜⎜
⎜
⎜
= ⎜1 − ⎜
⎟ ⎟ 1 − ⎜ ⎟⎟ ⎟⎟⎜ 1 − ⎜⎜ ⎟⎟ ⎟⎟...
⎜⎝ π ⎠⎟ ⎟⎟⎜⎜⎜
⎜⎜
⎝⎜ 3π ⎠ ⎠⎝
⎝⎜ 5π ⎠ ⎠⎟
⎟⎜⎜
⎝
⎠⎝
8.2
Geometric Mean Derivative of
tan x =
8x
π2 − (2x )2
+
8x
(3π)2 − (2x )2
41
cosx
+
8x
(5π)2 − (2x )2
+ ...
Gauge Institute Journal, Volume 4, No 4, November 2008
Proof:
D
(0)
e
cos x
−
sin x
cos x
= D
H. Vic Dannon
⎛
⎛ 2x ⎞⎟2 ⎞⎟⎟ (0) ⎛⎜
⎛ 2x ⎞⎟2 ⎞⎟⎟ (0) ⎛⎜
⎛ 2x ⎞⎟2 ⎞⎟⎟
⎜⎜ 1 − ⎜⎜⎜ ⎟⎟ ⎟⎟ D ⎜⎜⎜ 1 − ⎜⎜⎜ ⎟⎟ ⎟⎟ D ⎜⎜⎜ 1 − ⎜⎜⎜ ⎟⎟ ⎟⎟ ...
⎝ π ⎠ ⎠⎟
⎝ 3π ⎠ ⎠⎟
⎝ 5π ⎠ ⎠⎟
⎝⎜
⎝⎜
⎝⎜
(0) ⎜
⎜
=e
−
8x
2
2
π −(2x )
e
8x
8x
−
−
2
2
2
(3 π ) −(2x )
(5 π ) −(2x )2
e
...
Thus,
tan x =
8x
π 2 − (2x )2
+
8x
(3π)2 − (2x )2
42
+
8x
(5π)2 − (2x )2
+ ...
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
9
Product Calculus of
tan x
Product Representation for tan z
9.1
For any complex number z ,
tan z =
⎛
z 2 ⎞⎛
z 2 ⎞⎛
z 2 ⎞⎟
⎟
⎟
⎜
⎜
⎜
⎟⎟⎜ 1 −
⎟⎟ ...
z ⎜⎜ 1 − 2 ⎟⎟ ⎜⎜ 1 −
⎟⎜
⎟⎜⎜
⎜⎝
π ⎠⎝
(2π)2 ⎠⎝
(3π)2 ⎠⎟
⎛
⎛ 2z ⎞⎟2 ⎞⎛
⎛ 2z ⎞⎟2 ⎞⎛
⎛ 2z ⎞⎟2 ⎞⎟⎟
⎟⎟ ⎜⎜
⎟⎟ ⎜⎜
⎜⎜
⎜
⎜
⎜⎜ 1 − ⎜⎜ ⎟⎟ ⎟⎟⎜⎜ 1 − ⎜⎜ ⎟⎟ ⎟⎟⎜⎜ 1 − ⎜⎜⎜ ⎟⎟ ⎟⎟ ...
⎝ π ⎠ ⎠⎝
⎝ 3π ⎠ ⎠⎝
⎝ 5π ⎠ ⎠⎟
⎟⎜
⎟⎜
⎝⎜
The convergence is absolute in any disk z < R .
Proof: 7.1 and 8.1
9.2
Geometric Mean Derivative of
2
sin 2x
=
tanx
1
2x
8x
− 2
+
x π − x 2 π2 − (2x )2
−
−
2x
(2π)2 − x 2
2x
(3π)2 − x 2
+
+
Proof:
43
8x
(3π)2 − (2x )2
8x
(5π)2 − (2x )2
− ...
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
⎛
⎛ x ⎞⎟2 ⎞⎟⎟ (0) ⎛⎜
⎛ x ⎞⎟2 ⎞⎟⎟ (0) ⎛⎜
⎛ x ⎞⎟2 ⎞⎟⎟
⎜
⎜
D xD ⎜ 1 − ⎜ ⎟ ⎟⎟ D ⎜⎜ 1 − ⎜ ⎟ ⎟⎟ D ⎜⎜ 1 − ⎜⎜ ⎟ ⎟⎟...
⎜⎝ π ⎠⎟ ⎟
⎝⎜ 2π ⎠⎟ ⎠⎟
⎝⎜ 3π ⎠⎟ ⎠⎟
⎜⎝⎜
⎠
⎝⎜⎜
⎝⎜⎜
(0)
D tan x =
⎛
⎛ 2x ⎞⎟2 ⎞⎟⎟ (0) ⎛⎜
⎛ 2x ⎞⎟2 ⎞⎟⎟ (0) ⎛⎜
⎛ 2x ⎞⎟2 ⎞⎟⎟
⎜
(0) ⎜
D ⎜ 1 − ⎜⎜ ⎟ ⎟⎟ D ⎜⎜ 1 − ⎜⎜ ⎟ ⎟⎟ D ⎜⎜ 1 − ⎜⎜ ⎟ ⎟⎟ ...
⎜⎝ π ⎠⎟ ⎟
⎜⎝ 5π ⎠⎟ ⎟
⎜⎝⎜
⎜⎜⎝
⎝⎜ 3π ⎠⎟ ⎠⎟
⎠
⎝⎜⎜
⎠
(0)
e
2
sin 2 x
(0) ⎜
⎜
=
1
x
ee
e
−
−
2x
2
π −x 2
8x
2
π −(2 x )2
e
e
−
−
2x
(2 π )2 −x 2
e
8x
2
(3 π ) −(2 x )2
−
e
2x
(3 π )2 −x 2
−
8x
2
(5 π ) −(2 x )2
Thus,
2
sin 2x
=
1
2x
8x
− 2
+
x π − x 2 π 2 − (2x )2
−
−
2x
(2π)2 − x 2
2x
(3π)2 − x 2
+
+
44
....
8x
(3π)2 − (2x )2
8x
(5π)2 − (2x )2
− ...
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
10
Product Calculus of
sinh x
10.1 Product Representation of sinh z
⎛
z 2 ⎞⎛
z 2 ⎞⎛
z 2 ⎞⎟
⎟
⎟
⎜
⎜
⎜
sinh z = z ⎜⎜ 1 + 2 ⎟⎟⎜⎜ 1 +
⎟⎟⎟ ⎜⎜⎜ 1 +
⎟⎟ ....
⎟⎜
(2π)2 ⎠⎝
(3π)2 ⎠⎟
π ⎠⎝
⎝⎜
The convergence is absolute in any disk z < R .
Proof:
10.2
sinh z = −i sin iz , and use 7.1
Geometric Mean Derivative of
coth x =
Proof: D
(0)
sinhx
1
2x
2x
2x
+ 2
+
+
+ ...
x
π + x 2 (2π)2 + x 2 (3π)2 + x 2
⎛
x 2 ⎞⎟ (0) ⎛⎜
x 2 ⎞⎟ (0) ⎛⎜
x 2 ⎞⎟
⎜
sinh x = D xD ⎜ 1 +
⎟⎟ D ⎜ 1 +
⎟⎟ D ⎜ 1 +
⎟⎟ ....
⎜⎝
⎝⎜
⎝⎜
π 2 ⎠⎟
(2π)2 ⎠⎟
(3π)2 ⎠⎟
e
(0)
D sinh x
sinh x
(0)
1
x
=e e
2x
π2 + x 2
e
2x
(2 π )2 +x 2
e
2x
(3 π )2 +x 2
...
Thus,
coth x =
1
2x
2x
2x
+ 2
+
+
+ ...
2
2
2
x
(2π) + x
(3π)2 + x 2
π +x
45
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
11
Product Calculus of
cosh x
Product Representation of cosh z
11.1
2 ⎞⎛
2⎞
⎛
⎛ 2z ⎞⎟2 ⎞⎛
⎛
⎞
⎛
⎞
⎟
⎟
⎟
2
z
2
z
⎜⎜
⎜
⎜
cosh z = ⎜ 1 + ⎜⎜ ⎟⎟ ⎟⎟⎟⎜⎜ 1 + ⎜⎜ ⎟⎟⎟ ⎟⎟⎟⎜⎜ 1 + ⎜⎜ ⎟⎟⎟ ⎟⎟⎟ ....
⎜⎝ π ⎠ ⎟ ⎜⎜
⎜⎜
⎝⎜ 3π ⎠ ⎠⎝
⎝⎜ 5π ⎠ ⎠⎟
⎟ ⎜⎜
⎝
⎠⎝
The convergence is absolute in any disk z < R .
Proof: cosh z = cos iz , and apply 8.1
coshx
11.2 Geometric Mean Derivative of
tanh x =
Proof:
D
(0)
8x
π2 + (2x )2
+
8x
(3π)2 + (2x )2
+
8x
(5π)2 + (2x )2
+ ...
2
2
2
⎛
⎛ 2x ⎞⎟ ⎞⎟⎟ (0) ⎛⎜
⎛ 2x ⎞⎟ ⎞⎟⎟ (0) ⎛⎜
⎛ 2x ⎞⎟ ⎞⎟⎟
⎜⎜
⎜
⎜
⎜
cosh x = D ⎜ 1 + ⎜ ⎟ ⎟ D ⎜⎜ 1 + ⎜ ⎟ ⎟ D ⎜⎜ 1 + ⎜ ⎟ ⎟ ....
⎝⎜ π ⎠⎟ ⎠⎟⎟
⎝⎜ 3π ⎠⎟ ⎠⎟⎟
⎝⎜ 5π ⎠⎟ ⎠⎟⎟
⎝⎜
⎝⎜
⎝⎜
(0)
8x
8x
8x
D cosh x
2 +(2x )2 (3π)2 +(2x )2 (5π )2 +(2x )2
π
e cosh x = e
e
e
....
Thus,
tanh x =
8x
π 2 + (2x )2
+
8x
(3π)2 + (2x )2
46
+
8x
(5π)2 + (2x )2
+ ...
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
12
Product Calculus of
12.1
tanh x
Product Representation for tanh z
For any complex number z ,
⎛
z 2 ⎞⎛
z 2 ⎞⎛
z 2 ⎞⎟
⎟
⎟
⎜
⎜
⎜
⎟⎟ ⎜ 1 +
⎟⎟...
z ⎜⎜ 1 + 2 ⎟⎟ ⎜⎜ 1 +
⎟⎜
⎟ ⎜⎜
⎜⎝
(2π)2 ⎠⎝
(3π)2 ⎠⎟
π ⎠⎝
tanh z =
⎛
⎛ 2z ⎞⎟2 ⎞⎛
⎛ 2z ⎞⎟2 ⎞⎛
⎛ 2z ⎞⎟2 ⎞⎟⎟
⎟⎟ ⎜⎜
⎟⎟⎜⎜
⎜⎜
⎜
⎜
⎜⎜ 1 + ⎜⎜ ⎟⎟ ⎟⎟ ⎜⎜ 1 + ⎜⎜ ⎟⎟ ⎟⎟⎜⎜ 1 + ⎜⎜⎜ ⎟⎟ ⎟⎟...
⎝ π ⎠ ⎠⎝
⎝ 3π ⎠ ⎠⎝
⎝ 5π ⎠ ⎠⎟
⎟⎜
⎟⎜
⎝⎜
The convergence is absolute in any disk z < R .
Proof:
12.2
10.1 and 11.1
Geometric Mean Derivative of
2
sinh 2x
=
tanhx
1
2x
8x
+ 2
− 2
2
x
π +x
π + (2x )2
+
+
2x
2
(2π) + x
2
2x
2
(3π) + x
2
−
−
Proof:
47
8x
2
(3π) + (2x )2
8x
2
(5π) + (2x )2
+ ...
Gauge Institute Journal, Volume 4, No 4, November 2008
2
2
2
⎛
⎛ x ⎞⎟ ⎞⎟ (0) ⎛⎜
⎛ x ⎞⎟ ⎞⎟ (0) ⎛⎜
⎛ x ⎞⎟ ⎞⎟
⎜
⎟ D ⎜ 1 + ⎜⎜ ⎟ ⎟⎟ ...
D xD ⎜⎜ 1 + ⎜⎜ ⎟ ⎟⎟ D ⎜⎜ 1 + ⎜⎜
⎜
⎜⎝ π ⎠⎟ ⎟⎟
⎜ 2π ⎠⎟⎟ ⎟⎟⎟
⎝
⎝⎜ 3π ⎠⎟ ⎠⎟⎟
⎜⎝
⎜
⎠
⎝
⎠
⎝⎜
2
2⎞
2 ⎞
⎛
⎛
⎛ 2x ⎞⎟ ⎞⎟ (0) ⎛⎜
⎛
⎛
2x ⎞ ⎟ (0) ⎜
2x ⎞ ⎟
(0) ⎜
⎟
⎜ ⎟⎟
⎜ ⎟⎟
⎜
⎜
D ⎜⎜ 1 + ⎜⎜
⎜⎝ π ⎠⎟⎟ ⎟⎟⎟ D ⎜⎜ 1 + ⎝⎜ 3π ⎠⎟⎟ ⎟⎟⎟ D ⎜⎜ 1 + ⎝⎜ 5π ⎠⎟⎟ ⎟⎟⎟ ...
⎜⎝
⎝
⎠
⎠
⎝
⎠
(0)
D
(0)
tanh x
=
H. Vic Dannon
(0)
Now,
cosh2 x − sinh2 x
D tanh x
1
2
cosh2 x
=
=
=
.
tanh x
sinh x
sinh x cosh x
sinh 2x
cosh x
Therefore,
e
2
sinh 2 x
1
x
ee
=
e
2x
2x
2x
2
2
(3 π )2 +x 2
π2 +x 2 (2 π ) +x
e
e
8x
8x
8x
2
2
2
2
2
π +(2 x ) (3 π ) +(2 x ) (5 π ) +(2 x )2
e
e
Thus,
2
sinh 2x
=
1
2x
8x
+ 2
−
x
π + x 2 π 2 + (2x )2
+
+
...
2x
(2π)2 + x 2
2x
(3π)2 + x 2
−
−
48
8x
(3π)2 + (2x )2
8x
(5π)2 + (2x )2
+ ...
...
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
13
Product Calculus of
ex
13.1 Product representation of e x
⎛
⎞n
⎜⎜ 1 + z ⎟⎟ → e z
⎜⎝
n ⎠⎟ n →∞
13.2 Geometric Mean Derivative of e x
D (0)e x = e
Proof:
D
n
n
⎛
⎡ (0) ⎛
⎞⎤
x ⎞⎟
x
⎟
⎜⎜ 1 + ⎟⎟ = ⎢⎢ D ⎜⎜⎜ 1 + ⎟⎟ ⎥⎥
n⎠
n ⎠⎦
⎝
⎝
⎣
(0) ⎜
⎡ D (1+nx ) ⎤ n
⎢ 1+ x ⎥
= ⎢⎢ e n ⎥⎥
⎢
⎥
⎢⎣
⎥⎦
⎡ n1
⎢ 1+ x
= ⎢⎢ e n
⎢
⎢⎣
⎤n
⎥
⎥
⎥
⎥
⎥⎦
1
=e
1+ nx
⎯⎯
⎯⎯
→e
n →∞
Thus, D (0)e x = e .
49
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
13.3 Geometric Mean Derivative of ee
x
D (0)ee = ee
Proof:
D
⎛
ex
⎜1 +
⎜⎝⎜
n
(0) ⎜
⎞⎟n
⎡
⎛
ex
(0)
⎜
⎢
⎟⎟ = D ⎜ 1 +
⎢
n
⎠⎟
⎝⎜⎜
⎣
x
x
n
⎞⎤
⎟⎟ ⎥
⎟
⎠⎟ ⎥⎦
⎡ D (1+enx ) ⎤ n
⎢
⎥
⎢ 1+enx ⎥
= ⎢e
⎥
⎢
⎥
⎢
⎥
⎣⎢
⎦⎥
⎡ enx
⎢
⎢ 1+enx
= ⎢e
⎢
⎢
⎢⎣
⎤n
⎥
⎥
⎥
⎥
⎥
⎥⎦
ex
=e
1+e
→ ee
x
n
x
n →∞
x
x
Thus, D (0)ee = ee .
13.4 Geometric Mean Derivative of e x
k
D (0)e x = e kx
50
k −1
k
Gauge Institute Journal, Volume 4, No 4, November 2008
Proof:
D
⎛
xk
⎜⎜ 1 +
⎜⎝
n
(0) ⎜
⎞⎟n
⎡
⎛
xk
(0)
⎜
⎢
⎟⎟ = D ⎜ 1 +
⎢
n
⎠⎟
⎝⎜⎜
⎣
⎡ D (1+ xnk ) ⎤ n
⎢
⎥
⎢ 1+ xnk ⎥
= ⎢e
⎥
⎢
⎥
⎢
⎥
⎢⎣
⎥⎦
⎡ k x kn−1
⎢
⎢ 1+ xnk
= ⎢e
⎢
⎢
⎢⎣
⎤n
⎥
⎥
⎥
⎥
⎥
⎥⎦
kx k −1
=e
1+ x
→ e kx
k
n
k −1
n →∞
k
Thus, D (0)e x = e kx
k −1
.
51
H. Vic Dannon
n
⎞⎤
⎟⎟ ⎥
⎟
⎠⎟ ⎥⎦
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
14
Geometric
Mean
Derivative
by
Exponentiation
Geometric Mean Derivative can be obtained by using the Product
Calculus of the exponential function.
We demonstrate this method by examples.
D (0) sin x = e cot x
14.1
Proof: Since
sin x = e log sin x = lim (1 +
n →∞
log sin x n
) ,
n
we apply the Geometric Mean Derivative to (1 +
⎧⎛
⎛
log sin x ⎞⎟
log sin x ⎞⎫
⎪
⎪=
⎟⎟⎬⎪
D (0) ⎨⎪⎜⎜ 1 +
⎟⎟ × ... × ⎜⎜ 1 +
⎟
⎜
⎜
⎪
n
n
⎠
⎝
⎠⎪
⎪⎝
⎪
⎩
⎭
n factors
⎛
⎛
log sin x ⎞⎟
log sin x ⎞⎟
= D (0) ⎜⎜ 1 +
⎟⎟ × ... × D (0) ⎜⎜ 1 +
⎟
⎜⎝
n
n
⎠
⎝⎜
⎠⎟
n factors
52
log sin x n
)
n
.
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
⎡ D (1+ lognsin x ) ⎤ n
⎢
⎥
1+ log nsin x ⎥
⎢
= ⎢e
⎥
⎢
⎥
⎢⎣
⎥⎦
⎡ cotn x
⎢
1+ log nsin x
⎢
= ⎢e
⎢
⎢⎣
=e
⎤n
⎥
⎥
⎥
⎥
⎥⎦
cot x
1+ log sin x /n
→ e cotx .
n →∞
Thus, D (0) sin x = e cot x .
D (0)x x = ex
14.2
Proof:
Since
x n
x x = e x log x = lim (1 + x log
) ,
n
n →∞
x n
) .
we apply D (0) to (1 + x log
n
D
n
n
⎛
⎡ (0) ⎛
x log x ⎞⎟
x log x ⎞⎤
⎟
⎜
⎟ = ⎢ D ⎜1 +
⎟⎥
⎜⎜ 1 +
⎢⎣
n ⎠⎟
n ⎠⎟ ⎥⎦
⎝
⎝⎜
(0) ⎜
53
Gauge Institute Journal, Volume 4, No 4, November 2008
⎡ D (1+ x logn x ) ⎤ n
⎢
⎥
x
1+ x log
⎢
⎥
n
= ⎢e
⎥
⎢
⎥
⎢⎣
⎥⎦
x
⎡ 1+log
n
⎢
x
1+ x log
⎢
n
= ⎢e
⎢
⎢⎣
⎤n
⎥
⎥
⎥
⎥
⎥⎦
1+ log x
=e
x
1+ x log
n
→ e1+ log x = ex
n →∞
Thus, D (0)x x = ex .
54
H. Vic Dannon
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
15
Product Calculus of Γ(x )
On the half line x > 0 , Euler defined the real valued Gamma
function by
t =∞
Γ(x ) =
∫
e −tt x −1dt .
t =0
In the half plane Re z > 0 , the complex valued integral
t =∞
∫
e −tt z −1dt
t =0
converges, and is differentiable with
t =∞
Dz
∫
t =∞
−t z −1
e t
dt =
t =0
∫
e −tt z −1 ln tdt .
t =0
Thus, the complex valued integral extends the Euler integral into
an analytic function in the half plane Re z > 0 . It is denoted by
Γ(z ) .
This function can be further extended to a product representation
that is analytic for any z , except for simple poles that it has at
z = 0, −1, −2, −3,...
The function 1 / Γ(z ) , that is given by the inverse product, is
analytic for any z , with simple zeros at z = 0, −1, −2, −3,...
55
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Therefore, the natural calculus for Γ(z ) and for 1 / Γ(z ) in the
complex plane is the product Calculus.
Euler’s Product Representation for Γ(z )
15.1
z
z
z
( 1 + 11 ) ( 1 + 21 ) ( 1 + 13 ) ....
Γ(z ) =
z ( 1 + z1 )( 1 + z2 )( 1 + z3 ) ....
Proof:
t =∞
Γ(z ) =
∫
e −tt z −1dt
t =0
t =∞
=
∫
t =0
⎛
⎞⎟n z −1
t
lim ⎜⎜ 1 − ⎟⎟ t dt
⎜
n →∞ ⎝
n⎠
Uniform convergence allows order change of limit, and integration
t =∞
= lim
n →∞
∫
t =0
⎛
⎞n
⎜⎜ 1 − t ⎟⎟ t z −1dt
⎜⎝
n ⎠⎟
The change of variable, u = t / n , du = dt / n , gives
u =1
= lim n z
n →∞
n
∫ ( 1 − u ) u z −1du
u =0
n z can be written as a product
n z = (n + 1)z
z
nz
(n + 1)z
z
= ( 1 + 11 ) ( 1 + 21 ) ( 1 +
1
3
z
z
) ...( 1 + n1 )
56
nz
(n +1)z
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Integrating by parts with respect to u , keeping z , and n fixed
u =1
n
∫ (1 − u )
u =1
u
z −1
du =
u =0
n
∫ (1 − u )
u =0
⎛ uz
d ⎜⎜⎜
⎜⎝ z
⎞⎟
⎟⎟
⎠⎟
u =1
u =1 z
⎡ uz
n⎤
n
u
d (1 − u )
= ⎢ (1 − u ) ⎥
− ∫
⎢ z
⎥
⎣
⎦u = 0 u = 0 z
1
=
z
n
=
z
u =1
∫
n −1
u zn (1 − u )
du
u =0
u =1
∫ (1 − u )
u =0
n n −1
=
z z +1
⎛ u z +1 ⎞⎟
⎟
d ⎜⎜⎜
⎜⎝ z + 1 ⎠⎟⎟
n −1
u =1
∫ (1 − u )
u =0
n n −1n − 2
=
z z +1z +2
⎛ u z +2 ⎞⎟
⎟
d ⎜⎜⎜
⎜⎝ z + 2 ⎠⎟⎟
n −2
u =1
⎛ u z + 3 ⎞⎟
⎟
d ⎜⎜⎜
⎜⎝ z + 3 ⎠⎟⎟
n −3
∫ (1 − u )
u =0
…………………………………………………….
n n − 1 n − 2 n − (n − 1)
...
=
z z +1z + 2 z +n −1
u =1
u =0
u =1
n n − 1 n − 2 n − (n − 1) ⎛⎜ u z +n ⎞⎟
⎟
...
=
⎜
z z + 1 z + 2 z + n − 1 ⎜⎝⎜ z + n ⎠⎟⎟u = 0
57
⎛ u z +n ⎞⎟
d ⎜⎜⎜
⎟⎟
⎝⎜ z + n ⎠⎟
n −n
∫ (1 − u )
Gauge Institute Journal, Volume 4, No 4, November 2008
=
n n − 1 n − 2 n − (n − 1) 1
...
z z +1z + 2 z +n −1 z +n
=
n n − 1 n − 2 n − (n − 1) 1
...
z z +1z + 2 z +n −1 z +n
=
1 1
1
1
1
...
z ( z + 1 )( z + 1 ) ( z + 1 )( z + 1 )
1
2
n −1
n
H. Vic Dannon
Therefore,
u =1
lim n
n →∞
z
n
∫ (1 − u )
u z −1du =
u =0
= lim
n →∞
{( 1 +
1
1
z
z
z
z
) ( 1 + 21 ) ( 1 + 13 ) ...( 1 + n1 )
nz
(n +1)z
×
1 1
1
1
1 ⎫⎪⎪
...
×
⎬
z ( z + 1 )( z + 1 ) ( z + 1 )( z + 1 ) ⎪⎪
n −1
n
1
2
⎭
z
z
z
z
( 1 + 11 ) ( 1 + 21 ) ( 1 + 13 ) ...( 1 + n1 )
= lim
n →∞ z ( z + 1 )( z + 1 ) ... ( z + 1 )( z + 1 )
1
2
n −1
n
z
z
z
( 1 + 11 ) ( 1 + 21 ) ( 1 + 13 ) ....
=
.
z ( 1 + z1 )( 1 + z2 )( 1 + z3 ) ....
15.2
Geometric Mean Derivative of Γ(x )
⎛
Γ '(x )
1
1
1
1 ⎞⎟
= lim ⎜⎜ log n − −
−
− ... −
⎟
n →∞ ⎜
Γ(x )
x x +1 x +2
x + n ⎠⎟
⎝
58
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Proof:
x
D (0)Γ(x ) =
=
=
D (0) ( 1 + 11 ) D (0) ( 1 +
1
2
x
)
D (0) ( 1 +
1
3
x
)
....
D (0)xD (0) ( 1 + x1 ) D (0) ( 1 + x2 ) D (0) ( 1 + x3 ) ....
e
x
x
D 2x D (3/2) D (4/3)
x
x
(3/2)
(4/3)
2x
e
e
....
1
1
1 1
+
+
+
x
1
x
2
x
x
e e e e 3 ....
234
....
123
1
1
1
1
....
+
+
+
1
2
3
x
x
x
+
+
+
x
e
⎛ 2 3 4 ⎞⎟ − 1 − 1 − 1 − 1 ....
.... ⎟e x x +1 x +2 x + 3
= ⎜⎜
⎜⎝ 1 2 3 ⎠⎟
= lim (n
n →∞
1
1
1
1
.....−
−
− −
1
2
+
+
+
x
x
x
n
x
+ 1)e
= lim
1
1
1
1
.....−
−
− −
1
2
+
+
+
x
x
x
n
x
ne
= lim
1
1
1
1
−
−....−
log n − −
1
2
+
+
+
x
x
x
n
x
e
n →∞
n →∞
=e
⎛
1
1
1
1 ⎞⎟
lim ⎜⎜ log n − −
−
−.....−
⎟
⎜
n →∞⎝
x x +1 x + 2
x +n ⎠⎟
Γ '(x )
Since D(0)Γ(x ) = e Γ(x ) ,
59
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
⎛
1
1
1
1 ⎞⎟
Γ '(x )
= lim ⎜⎜ log n − −
−
− ... −
⎟
n →∞ ⎜
Γ(x )
x x +1 x +2
x + n ⎠⎟
⎝
Γ(1) = 1
15.3
Proof:
( 1 + 11 )( 1 + 21 )( 1 + 13 )....
Γ(1) =
=1
1( 1 + 11 )( 1 + 21 )( 1 + 13 ) ....
Γ(z + 1) = z Γ(z )
15.4
Proof:
z +1
Γ(z + 1) =
z +1
....
z ( 1 + z +1 1 )( 1 + z +2 1 )( 1 + z +3 1 )....
z
=
z +1
( 1 + 11 ) ( 1 + 21 ) ( 1 + 13 )
z
z
( 1 + 11 ) 2 ( 1 + 21 ) 23 ( 1 + 13 )
( z + 1)( 1 + z +1 1 )( 1 + z +2 1 )( 1 +
z
z
4 ....
3
z +1
3
z
)....
( 1 + 11 ) ( 1 + 21 ) ( 1 + 13 ) ...
=
( 1 + z ) 21 ( 2 + z ) 23 ( 23 + z2 ) 43 ( 43 + z3 )...
z
z
z
( 1 + 11 ) ( 1 + 21 ) ( 1 + 13 ) ...
=
( 1 + z )( 1 + z2 )( 1 + z3 )( 1 + z4 )...
= z Γ(z ) .
60
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Product Reflection Formula for Γ(z )
15.5
Γ(z )Γ(1 − z ) =
1
(
)( 1 − z3 )...
2
z (1 − z 2 ) 1 − z2
2
2
2
Proof:
z
z
1−z
z
1−z
1−z
( 1 + 11 ) ( 1 + 21 ) ( 1 + 13 ) ... ( 1 + 11 ) ( 1 + 21 ) ( 1 + 13 ) ...
Γ(z )Γ(1 − z ) =
z ( 1 + z1 )( 1 + z2 )( 1 + z3 ) ... ( 1 − z ) ( 1 + 1−1 z )( 1 + 1−2 z )( 1 + 1−3 z ) ..
( 1 + 11 )( 1 + 21 )( 1 + 13 )...
=
z ( 1 + z )( 1 + z )( 1 + z ) .. ( 1 − z ) 2 ( 1 − z ) 3 ( 1 − z ) 4 ( 1 − z )..
2
3
2 2
3 3
4
=
=
=
1
z ( 1 + z1 )( 1 + z2 )( 1 + z3 ) .. ( 1 − z ) ( 1 − z2 )( 1 − z3 )( 1 − z4 ) ..
1
z ( 1 + z )( 1 − z ) ( 1 + z2 )( 1 − z2 )( 1 +
1
(
z (1 − z 2 ) 1 − z2
2
2
)(
2
3
Γ(z )Γ(1 − z ) =
15.6
)
1 − z 2 ...
.
π
sin πz
Proof: By 15.5,
Γ(z )Γ(1 − z ) =
(
1
)( 1 − z3 )...
2
z (1 − z 2 ) 1 − z2
2
2
2
61
z
3
)( 1 − z3 )..
Gauge Institute Journal, Volume 4, No 4, November 2008
=
=
(
2
πz 1 − (πz2)
π
)(
π
2
1 − (πz )2
(2 π )
)(
H. Vic Dannon
2
)
1 − (πz )2 ..
(3 π )
π
.
sin πz
2
( Γ(21) )
15.7
Proof: Substituting z =
1
2
=π
in Γ(z )Γ(1 − z ) =
Γ ( 21 ) Γ ( 1 − 21 ) =
π
, we obtain
sin πz
π
.
sin π2
That is,
⎡ Γ( 1) ⎤ 2 = π .
⎢⎣ 2 ⎥⎦
This can be obtained directly through the Wallis Product for π .
⎡ Γ ( 1 ) ⎤ 2 = 2 ⋅ 2 2 4 4 6 6 8 8 10 10 12 12 ...
⎢⎣ 2 ⎥⎦
1 3 3 5 5 7 7 9 9 11 11 13
15.8
Proof: By 15.5,
1
⎡ Γ ( 1 )⎤2 =
⎢⎣ 2 ⎥⎦
1 ⎛⎜
1 ⎞⎛
1 ⎞⎛
1 ⎞⎛
1 ⎞⎛
1 ⎞⎟
⎟
⎟
⎟
⎟
⎜
⎜
⎜
⎜
1
−
1
−
1
−
1
−
1
−
⎟
⎟
⎟
⎟
⎟...
⎜
⎟⎜⎜
⎟ ⎜⎜
⎟ ⎜⎜
⎟ ⎜⎜
2 ⎜⎝
22 ⎠⎝
42 ⎠⎝
62 ⎠⎝
82 ⎠⎝
102 ⎠⎟
=
1
1 ⎛⎜ 1 ⋅ 3 ⎞⎛
3 ⋅ 5 ⎞⎛ 5 ⋅ 7 ⎞⎛ 7 ⋅ 9 ⎞⎛ 9 ⋅ 11 ⎞
⎜⎜ 2 ⎟⎟⎟ ⎜⎜⎜ 2 ⎟⎟⎟ ⎜⎜⎜ 2 ⎟⎟⎟⎜⎜⎜ 2 ⎟⎟⎟⎜⎜⎜ 2 ⎟⎟⎟ ...
2 ⎝ 2 ⎠⎝ 4 ⎠⎝ 6 ⎠⎝ 8 ⎠⎝ 10 ⎠
62
Gauge Institute Journal, Volume 4, No 4, November 2008
⎛ 2 2 ⎞⎛ 4 4 ⎞⎛ 6 6 ⎞⎛ 8 8 ⎞⎛ 10 10 ⎞
= 2 ⋅ ⎜⎜ ⋅ ⎟⎟ ⎜⎜ ⋅ ⎟⎟ ⎜⎜ ⋅ ⎟⎟⎜⎜ ⋅ ⎟⎟ ⎜⎜ ⋅ ⎟⎟ ....
⎟ ⎜ 3 5 ⎟⎠⎝⎜ 5 7 ⎠⎝
⎟⎜ 7 9 ⎠⎝
⎟ ⎜ 9 11 ⎠⎟
⎜⎝ 1 3 ⎠⎝
Wallis Formula of 7.3, follows from 15.7, and 15.8.
63
H. Vic Dannon
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
16
Products of Γ(z )
Γ(1 + z1 )
, where z1 = w1 + w2
Γ(1 + w1 )Γ(1 + w2 )
16.1
(
w
)(
) × (1 + )(1 + ) × ...
)
(1 + )
1
( 1 + w1 )( 1 + w2 ) 1 + 2 1 +
Γ(1 + z1 )
=
×
z
Γ(1 + w1 )Γ(1 + w2 )
( 1 + z1 )
1 + 21
(
w2
2
w1
3
z1
3
Proof:
Γ(1 + z1)
=
Γ(1 + w1)Γ(1 + w2 )
1+ z1
=
( 1 + 11 )
( 1 + z1 ) ( 1 +
1+ z 1
( 1 + 21 )
1+ z 1
1
)(1 +
( 1 + w1 )( 1 + 1+1w
1
×
×
1+w1
( 1 + 11 )
1+w2
)(1 +
)(1 +
1+w1
2
1+w1
2
)(1 +
1+w2
2
(
z1
2
)(
1+
z1
z1
3
)(
)(1 +
)(1 +
1+
64
....
z1
4
)
....
×
×
)....
1+w1
3
)... ×
...
1+w2
3
1+w2
( 1 + 13 )
( 1 + 11 ) ( 1 + 21 ) ( 1 + 13 )
( 1 + z1 ) 1 +
1+ z1
3
1+w1
1+w2
z1
....
( 1 + 13 )
( 1 + 21 )
z1
=
1+ z1
2
( 1 + 21 )
( 1 + w2 )( 1 + 1+1w
( 1 + 11 )
1+ z 1
( 1 + 13 )
w2
3
...
)... .
Gauge Institute Journal, Volume 4, No 4, November 2008
( 1 + w1 )( 1 + w2 )( 1 + w3 )( 1 + w4 )...
1
×
1
w1
1
2
w1
1
3
w1
×
...
( 1 + w2 )( 1 + w2 )( 1 + w3 )( 1 + w4 )...
2
2
w2
2
w2
w2
(1 + ) (1 + ) (1 + )
1
1
1
2
1
3
.
...
( 1 + w1 )( 1 + w2 )( 1 + w3 )...( 1 + w2 )( 1 + w2 )( 1 + w3 )...
1
=
1
(1 + ) (1 + ) (1 + )
1
1
×
H. Vic Dannon
1
2
( 1 + z1 )( 1 + z2 )( 1 + z3 )( 1 + z4 )...
1
1
( 1 + w1 )( 1 + w2 ) ( 1 + 2 )( 1 +
=
×
( 1 + z1 )
(1 + z2 )
w1
1
w2
2
1
16.2
2
) × (1 + )(1 + ) × ...
(1 + )
w1
3
w2
3
z1
3
Γ(1)
= sinh x
Γ(1 + ix )Γ(1 − ix )
Proof:
Γ(1)
= ( 1 + ix )( 1 − ix ) ( 1 + ix2 )( 1 − ix2 )( 1 + ix3 )( 1 − ix3 )...
Γ(1 + ix )Γ(1 − ix )
(
= (1 + x 2 ) 1 +
x2
22
= sinh x .
Similarly, we obtain
65
)(1 + x3 )...
2
2
Gauge Institute Journal, Volume 4, No 4, November 2008
16.3
H. Vic Dannon
Γ(1 + z1)Γ(1 + z 2 )
, where z1 + z 2 = w1 + w2 + w 3 .
Γ(1 + w1)Γ(1 + w2 )Γ(1 + w 3 )
Γ(1 + z1)Γ(1 + z 2 )
=
Γ(1 + w1)Γ(1 + w2 )Γ(1 + w 3 )
( 1 + w1 )( 1 + w2 )( 1 + w3 ) ( 1 + 2 )( 1 + 2 )( 1 +
=
×
( 1 + z1 )( 1 + z 2 )
(1 + z2 )(1 + z2 )
w1
w2
1
w3
2
)×
2
(1 + )(1 + )(1 + ) × ...
×
(1 + )(1 + )
w1
3
w2
3
z1
3
w3
3
z2
3
More generally,
16.4
If z1 + z 2 + ... + z k = w1 + w2 + ... + wl ,
Γ(1 + z1)Γ(1 + z 2 )...Γ(1 + z k )
may be simplified
Γ(1 + w1)Γ(1 + w2 )....Γ(1 + wl )
A weaker result that requires that k = l , is stated in [Melz, p.
101], and in [Rain, p. 249].
66
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
17
Product Calculus of
For
a complex number ν ,
J ν(x )
the Bessel function J ν (z ) solves
Bessel’s differential equation
z
2d
2
w
dz 2
+z
dw
+ (z 2 − ν 2 )w = 0 .
dz
For ν real, J ν (z ) has infinitely many real zeros, all simple with
the possible exception of z = 0 .
For ν ≥ 0 , the positive zeros
jν,k are a monotonic increasing
sequence
jν,1 < jν,2 < jν,3 < ...
17.1 Product Formula for J ν (z ) [Abram, p.370]
⎛
⎛ z ⎞⎟ν
1
z 2 ⎞⎛
z 2 ⎞⎛
z 2 ⎞⎟⎟
⎟⎟ ⎜⎜
⎟⎟⎜⎜
⎜⎜
⎜
J ν (z ) = ⎜ ⎟⎟
1 − 2 ⎟⎟ ⎜ 1 − 2 ⎟⎟⎜ 1 − 2 ⎟⎟ ...
⎜⎝ 2 ⎠ Γ(ν + 1) ⎜⎜
⎟⎜
⎟⎜
jν,1 ⎠⎝
jν,2 ⎠⎝
jν,3 ⎠⎟
⎝
17.2
Geometric Mean Derivative of J ν (z )
DJ ν (x ) ν
2x
2x
2x
= − 2
−
−
− ...
J ν (x )
x
jν,1 − x 2 jν2,2 − x 2 jν2,3 − x 2
67
Gauge Institute Journal, Volume 4, No 4, November 2008
Proof: D (0)J ν (x ) = D (0)
1
ν
2 Γ(ν + 1)
H. Vic Dannon
D (0)x ν ×
⎛
⎛
⎛
x 2 ⎞⎟
x 2 ⎞⎟
x 2 ⎞⎟
⎜
⎜
⎜
×D (0) ⎜⎜ 1 − 2 ⎟⎟⎟ D (0) ⎜⎜ 1 − 2 ⎟⎟⎟ D (0) ⎜⎜ 1 − 2 ⎟⎟⎟...
⎜⎝
jν,1 ⎠⎟
jν,2 ⎠⎟
jν,3 ⎠⎟
⎝⎜
⎝⎜
= e 0e
Dx ν
xν
e
D(1−x 2 / jν2,1) D(1−x 2 / jν2,2 ) D(1−x 2 / jν2,3 )
1−x 2 / jν2,1
1−x 2 / jν2,2
1−x 2 / jν2,3
e
e
−2x
−2x
−2x
ν 2
2
2
2
2
jν ,1−x jν ,2 −x jν ,3 −x 2
x
=e e
e
e
...
2x
2x
2x
−
−
ν − 2
jν ,1−x 2 jν2,1−x 2 jν2,2 −x 2
x
=e e
...
Thus,
DJ ν (x ) ν
2x
2x
2x
= − 2
−
−
− ...
J ν (x )
x
jν,1 − x 2 jν2,2 − x 2 jν2,3 − x 2
68
...
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
18
Product Calculus of Trigonometric
Series
If
∞
⎛
1
n πx
n πx ⎞⎟
f (x ) = a 0 + ∑ ⎜⎜an cos
+ bn sin
⎟
⎜
L
L ⎠⎟
2
n =1 ⎝
the Geometric Mean Derivative can be applied to
e
⎛
⎞
⎜⎜ f (x )− 1a ⎟⎟
2 0 ⎠⎟
⎝⎜
=
e
⎛
⎞
⎜⎜a cos πx +b sin πx ⎟⎟
1
L
L ⎠⎟
⎝⎜ 1
e
⎛
⎞
⎜⎜a cos 2 πx +b sin 2πx ⎟⎟
2
L
L ⎠⎟
⎝⎜ 2
....
18.1 Product Integral of a Trigonometric Series
on [0,2π ] ,
⎛ sin x
⎞
sin 2x
sin 3x
+
+
+ ... ⎟⎟⎟
x = π − 2 ⎜⎜
⎜⎝ 1
2
3
⎠
Therefore,
e
x −π
=e
−2 sin x
e
−2
sin 2x
sin 3x
−2
2 e
3 .... .
Product Integrating both sides,
e
⎛1 2
⎞
⎜⎜ x − πx ⎟⎟
⎜⎝ 2
⎠⎟
=e
⎛ cos 2x ⎞⎟
⎜⎜ 2
⎟
2 cos x ⎝⎜ 22 ⎠⎟
e
e
⎛ cos 3x ⎞⎟
⎜⎜ 2
⎟
⎝⎜ 32 ⎠⎟
....
Hence,
1 2
cos 2x
cos 3x
x − πx = 2 cos x + 2 2 + 2 2 + ....
2
2
3
69
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
19
Infinite Functional Products
Euler represented Analytic functions by infinite products, [Saks].
to which the Geometric Mean derivative may be applied.
19.1 Geometric Mean Derivative of an Euler Product
Consider Euler’s product
2
1
= ( 1 + x )( 1 + x 2 ) 1 + x 2
1−x
(
)(1 + x ).... .
23
Applying the Geometric Mean Derivative to both sides,
e
⎛ 1 ⎞⎟
⎜⎜
⎟
⎜⎝ 1−x ⎠⎟
2
=
3
n
2 −1 22 x 2 −1 23 x 2 −1
2n x 2 −1
1 2x
2
22
23
2n
e 1+x e 1+x e 1+x e 1+x ....e 1+x ...
Hence,
1
2
n
1
1
21x 2 −1 22 x 2 −1
2n x 2 −1
=
+
+
+ ...
+ ...
n
2
1−x
1 + x 1 + x 21
1 + x2
1 + x2
70
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
20
Path Product Integral
20.1 Path Product Integral in the plane
Let P (x, y ) , and Q(x , y ) be positive, and smooth so that
log P (x , y ) is integrable with respect to x ,
and
logQ(x , y ) is integrable with respect to y ,
along the path γ in the x , y Plane, from (x1, y1) to (x 2, y2 ) .
We define the Product Integral along the path γ by
dx
dy
∏( P(x, y )) (Q(x, y ) )
γ
≡
dx
dy
∏( P(x, y ) ) ∏(Q(x, y ))
γ
γ
∫ ( log P (x ,y ))dx ∫ ( log Q (x ,y ))dy
eγ
=eγ
∫ ( log P (x ,y ))dx +( log Q (x ,y ))dy
=eγ
20.2 Green’s Theorem for the Path Product Integral
Path Product Integral over a loop equals
e
Proof:
⎛ ∂y P ∂ Q ⎞⎟
⎜
∫∫ ⎜⎜⎜⎝ P − Qx ⎠⎟⎟⎟dxdy
inerior ( γ )
By Green’s Theorem.
71
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
20.3 Path Product Integral in E 3
If γ is a path on a smooth surface in three dimensional space,
we define the path product Integral along γ by
dx
dy
dz
∏( P(x, y, z )) (Q(x, y, z )) ( R(x, y, z ))
∫ ( log P (x ,y,z ))dx +( logQ (x ,y,z ))dy +( log R(x ,y,z ) )dz
≡eγ
γ
20.4
Stokes Theorem for the Path Product Integral
path product Integral over a loop in a smooth surface in E 3
equals
e
∫∫
⎡
⎤ ⎛
⎞
⎢ log P (x ,y ,z ) ⎥ ⎜⎜ dydz ⎟⎟
⎟
∇×⎢⎢ log Q (x ,y ,z ) ⎥⎥ i⎜⎜⎜ dxdz ⎟⎟
⎟
⎢ log R(x ,y ,z ) ⎥ ⎜⎜⎜dxdy ⎟⎟⎟
⎠
⎣⎢
⎦⎥ ⎝
Proof: By Stokes Theorem.
72
.
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
21
Iterative Product Integral
21.1 Iterative Product Integral of f (x , t )
Let f (x , t ) be positive in the rectangle
[x 0, x1 ] × [t0, t1 ]
so that
log f (x , t )
is integrable on the rectangle.
Then, the double product integral is defined iteratively by
⎞⎟dt
⎞
⎜⎜ x =∫x0 log f (x ,t )dx ⎟⎟
⎜⎜
dx ⎟
⎟
⎟⎟
⎜⎜e
⎜⎜ ∏ f (x , t ) ⎟⎟⎟ =
⎟⎟
⎠
t =t 0 ⎝ x = x 0
t =t0 ⎜
⎜⎝
⎠⎟
t =t1 ⎛
⎜
dt
t =t1 ⎛ x = x1
x = x1
∏
∏
t =t1 x =x1
∫ ∫
log f (x ,t )dxdt
= e t =t0 x =x 0
21.2 Iterative Product Integral of e r (x ,t )dx
dt
t =t1 x =x1
∫ ∫ r (x ,t )dxdt
⎞⎟
⎜⎜
∏ ⎜⎜ ∏ er (x ,t )dx ⎟⎟⎟⎟ = et =t0 x =x0
⎠
t =t ⎝ x =x
t =t1 ⎛ x =x1
0
0
73
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
22
Harmonic Mean Integral
22.1
Harmonic Mean Integral
The electrical voltage on a capacitance C (q ) due to a charge dq at
x is
dq
dq(x )
dx
=
≡
,
C (q ) C (q(x ))
f (x )
where the function f (x ) is real and non-vanishing.
Over n equal sub-intervals of the interval [a,b ] , Δx =
b −a
,
n
we obtain the sequence of voltages
⎛ 1
1
1
1
1
1 ⎞⎟
⎟⎟ Δx .
Δx +
Δx + ... +
Δx = ⎜⎜
+
+ ... +
⎜⎝ f (x ) f (x )
⎟
(
)
f (x1 )
f (x 2 )
f (x n )
f
x
⎠
1
2
n
As n → ∞ , the sequence converges to
x =b
∫
x =a
1
dx .
f (x )
We call the limit
the Harmonic Mean integral of f (x ) over the interval [a,b ] ,
and denote it by
x =b
(−1)
I
x =a
f (x ) .
The inverse operation to Harmonic Mean integration is the
Harmonic Mean Derivative
74
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
23
Harmonic Mean, and Harmonic Mean
Derivative
f (x ) over [a,b ]
23.1 The Harmonic Mean of
Given an non-vanishing integrable function f (x ) over the interval
[a,b ] , partition the interval into n sub-intervals, of equal length
Δx =
b −a
,
n
choose in each subinterval a point
ci ,
and consider the Harmonic Mean of f (x )
n
1
1
1
1
+
+ ... +
f (c1) f (c2 )
f (cn )
=
b −a
⎛ 1
1
1 ⎞⎟
+
+ ... +
⎟ Δx
⎜⎜⎜
f (cn ) ⎠⎟
⎝ f (c1 ) f (c2 )
As n → ∞ , the sequence of Harmonic Means converges to
b −a
x =b
∫
x =a
1
dx
f (x )
=
b −a
,
H (b) − H (a )
where
75
Gauge Institute Journal, Volume 4, No 4, November 2008
t =x
∫
H (x ) =
t =0
H. Vic Dannon
1
dt .
f (t )
Therefore,
b −a
x =b
∫
x =a
1
dx
f (x )
is defined as the Harmonic Mean of f (x ) over [a,b ] .
23.2
Mean Value Theorem for the Harmonic Mean
There is a point a < c < b , so that
b −a
x =b
∫
x =a
23.3
= f (c)
1
dx
f (x )
The Harmonic Mean of f (x ) over [x , x + dx ]
The Harmonic Mean at x is the Inverse operation to
Harmonic Mean Integration
Proof: The Harmonic Mean of f (x ) over the interval [x , x + Δx ] , is
Δx
t = x +Δx
∫
t =x
.
1
dt
f (t )
Letting Δx → 0 , the Harmonic Mean of f (x ) at x equals f (x ) .
76
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
Δx
lim
Δx → 0 t = x +Δx
∫
t =x
= f (x ) .
1
dt
f (t )
Thus, the operation of finding the Harmonic Mean of f (x ) at the
point x , is inverse to Harmonic Mean Integration.
This leads to the definition of the Harmonic Mean Derivative
23.4 Harmonic Mean Derivative
The Harmonic Mean Derivative of
t =x
H (x ) =
∫
t =0
1
dt
f (t )
at x is defined as the Harmonic Mean of f (x ) at x
Δx
D (−1)H (x ) ≡ lim
Δx → 0 t =x +Δx
∫
t =x
D (−1)H (x ) =
23.5
1
dt
f (t )
1
Dx H (x )
Proof:
D (−1)H (x ) = Standard Part of
=
1
.
Dx H (x )
77
dx
H (x + dx ) − H (x )
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
24
Harmonic Mean Calculus
24.1 The Fundamental Theorem of the Harmonic Mean
Calculus
Dx
Proof:
Dx
⎛ t =x
(−1) ⎜
⎜ (−1)
⎜⎜ I
⎜⎝ t =a
⎛ t =x
(−1) ⎜
⎜ (−1)
⎜⎜ I
⎜⎝ t =a
⎞⎟
f (t ) ⎟⎟⎟ = f (x ) .
⎠⎟⎟
⎛ t =x
⎞⎟
⎞⎟
1
(−1) ⎜
⎟
f (t ) ⎟⎟ = Dx
dt ⎟⎟⎟
⎜⎜⎜ ∫
⎟
⎠⎟⎟
⎝⎜ t =a f (t ) ⎠⎟
1
=
t =x
Dx
∫
t =a
1
dt
f (t )
= f (x ) .
24.2 Table of Harmonic Mean Derivatives and integrals
We list some Harmonic Mean Derivatives, and Integrals.
78
Gauge Institute Journal, Volume 4, No 4, November 2008
f (x )
x
D (−1) f (x )
x2
1
2x
1
ax a −1
xa
1
1
x
−x 2
log x
x
ex
e−x
e 2x
cos x
tan x
1 −2x
e
2
1
x x (log x + 1)
1
cos x
1
sin x
sin2 x
eex
e−xe−ex
xx
sin x
H. Vic Dannon
I (−1)f (x )
log x
−1
x
1
(a − 1)x a −1
1 2
x
2
1
∫ log x dx
−e−x
e sin x cos xe sin x
e cos x − sin xe cos x
79
− 1 e−2x
2
1
∫ x x dx
dx
∫ sin x
dx
∫ cos x
logsin x
1
∫ ee dx
∫ e− sin xdx
∫ e− cos xdx
x
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
25
Quadratic Mean Integral
25.1 Quadratic Mean Integral
Given a Riemann integrable, positive f (x ) on [a,b ] , partition the
interval into n sub-intervals, of equal length
Δx =
b −a
,
n
choose in each subinterval a point
ci ,
and consider the finite products,
( f 2(c1) + f 2(c2 ) + ... + f 2(cn ) ) Δx
As n → ∞ , the sequence converges to
x =b
∫
f 2 (x )dx .
x =a
We call this limit
the Quadratic Mean Integral of f (x ) over the interval [a,b ] ,
and denote it by
x =b
(2)
I
x =a
f (x ) = f
2
L2[a,b ]
Thus, Quadratic Mean integration transforms a function to its L2
norm squared.
80
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
25.2 Cauchy-Schwartz inequality for Quadratic Mean
Integrals
1
1
1
⎛ x =b
⎞⎟2 ⎛ x =b
⎞⎟2 ⎛ x =b
⎞⎟2
⎜⎜ (2) ⎡
⎜
⎜⎜ (2)
(2)
⎟
⎟
⎟
⎜
⎤
⎜⎜ I ⎣ f (x ) + g(x ) ⎦ ⎟⎟ ≤ ⎜⎜ I f (x ) ⎟⎟ + ⎜⎜ I g(x ) ⎟⎟
x
=
a
x
=
a
x
=
a
⎜⎝
⎠⎟⎟
⎝⎜
⎠⎟⎟
⎝⎜
⎠⎟⎟
Proof:
1
⎞2
⎛ x =b
⎜⎜ (2)
⎟⎟
⎜⎜ I ⎣⎡ f (x ) + g(x ) ⎦⎤ ⎟⎟
⎜⎝ x =a
⎠⎟⎟
1
⎛ x =b
⎞⎟2
⎜⎜
2
⎟
= ⎜ ∫ ( f (x ) + g(x ) ) dx ⎟⎟
⎟
⎜⎝⎜ x =a
⎠⎟
By Cauchy-Schwartz Inequality,
⎛ x =b
⎞⎟1/2 ⎛ x =b
⎞⎟1/2
⎜⎜
⎜
2
2
≤ ⎜ ∫ ( f (x ) ) dx ⎟⎟⎟ + ⎜⎜ ∫ ( g(x ) ) dx ⎟⎟⎟
⎜⎜
⎜
⎟
⎟
⎝ x =a
⎠⎟
⎝⎜ x =a
⎠⎟
⎛ x =b
⎜⎜ (2)
1
⎞2
1
⎞2
⎛ x =b
⎜⎜ (2)
⎟
⎟
= ⎜ I f (x ) ⎟⎟⎟ + ⎜ I g(x ) ⎟⎟⎟ .
⎜⎜ x =a
⎜ x =a
⎝
⎠⎟⎟
⎝⎜
⎠⎟⎟
25.3
Holder Inequality for Quadratic Mean Integrals
x =b
∫
x =a
Proof:
1
1
⎛ x =b
⎞⎟2 ⎛ x =b
⎞⎟2
⎜⎜ (2)
⎜⎜ (2)
⎟
f (x )g(x ) dx ≤ ⎜ I f (x ) ⎟⎟ ⎜ I g(x ) ⎟⎟⎟
⎜⎜ x =a
⎜ x =a
⎝
⎠⎟⎟ ⎝⎜
⎠⎟⎟
By Holder’s inequality,
x =b
∫
x =a
⎛ x =b
⎞⎟1/2 ⎛ x =b
⎞⎟1/2
⎜⎜
⎜
f (x )g(x ) dx ≤ ⎜ ∫ f 2(x )dx ⎟⎟⎟ ⎜⎜ ∫ g 2(x )dx ⎟⎟⎟
⎜⎜
⎟ ⎜
⎟
⎝ x =a
⎠⎟ ⎝⎜ x =a
⎠⎟
81
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
1
1
⎛ x =b
⎞⎟2 ⎛ x =b
⎞⎟2
⎜⎜ (2)
⎜⎜ (2)
⎟
≤ ⎜ I f (x ) ⎟⎟ ⎜ I g(x ) ⎟⎟⎟ .
⎜⎜ x =a
⎜ x =a
⎝
⎠⎟⎟ ⎝⎜
⎠⎟⎟
82
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
26
Quadratic Mean and Quadratic Mean
Derivative
26.1
Quadratic Mean of f (x ) over [a,b ]
Given an integrable positive function f (x ) on [a,b ] , partition the
interval into n sub-intervals, of equal length Δx =
b −a
,
n
choose in each subinterval a point ci ,
and consider the Quadratic Means of f (x )
⎛ f 2 (c ) + f 2 (c ) + ... + f 2 (c ) ⎞⎟1/2 ⎛ 1
⎞⎟1/2
⎜⎜
2
2
2
1
2
n ⎟
⎜
( f (c1) + f (c2 ) + ... + f (cn ) ) Δx ⎠⎟⎟
⎟ = ⎜⎜
⎜⎜
n
⎝b − a
⎝
⎠⎟
As n → ∞ , the sequence of Quadratic Means converges to
x =b
⎛
⎞⎟1/2 ⎛
⎞⎟1/2
⎜⎜ 1
Q
(
b
)
Q
(
a
)
−
⎟
2
f (x )dx ⎟⎟ = ⎜⎜
⎟⎟ ,
⎜⎜
∫
⎜
⎟
−
−
b
a
b
a
⎝
⎠
⎜⎝
⎠⎟
x =a
where
t =x
Q(x ) =
∫
f 2 (t )dt .
t =0
x =b
⎛
⎞⎟1/2
⎜⎜ 1
f 2 (x )dx ⎟⎟⎟
⎜⎜
∫
⎟
⎜⎝ b − a x =a
⎠⎟
is defined as the Quadratic Mean of f (x ) over [a,b ] .
83
Gauge Institute Journal, Volume 4, No 4, November 2008
26.2
H. Vic Dannon
Mean Value theorem for the Quadratic Mean
1
x =b
⎛
⎞⎟2
⎜⎜ 1
There is a point a < c < b , so that ⎜ b −a ∫ f 2(x )dx ⎟⎟⎟ = f (c)
⎜⎜
⎟
⎝
⎠⎟
x =a
26.3 The Quadratic Mean of f (x ) over [x , x + dx ]
The Quadratic Mean at x , is the Inverse operation to
Quadratic Mean Integration
Proof: By 26.2, there is
x < c < x + Δx ,
so that
⎛ t =x +Δx
⎞⎟1/2
⎜⎜ 1
⎟
2
⎜⎜ Δx ∫ f (t )dt ⎟⎟ = f (c) .
⎟
⎜⎝
⎠⎟
t =x
Letting Δx → 0 , the Quadratic Mean of f (x ) at x equals f (x ) .
t =x +Δx
⎛
⎞⎟1/2
⎜ 1
lim ⎜⎜
f 2(t )dt ⎟⎟⎟ = f (x ) .
∫
⎜
Δx → 0 ⎜ Δx
⎟
⎝
⎠⎟
t =x
Thus, the operation of finding the Quadratic Mean of f (x ) at the
point x , is inverse to Quadratic Mean Integration.
This leads to the definition of Quadratic Mean Derivative
26.4 Quadratic Mean Derivative
84
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
The Quadratic Mean Derivative of
t =x
Q(x ) =
∫
f 2 (t )dt
t =0
at x is defined as the Quadratic Mean of f (x ) at x
t = x +Δx
⎛
⎞⎟1/2
⎜ 1
D (2)Q(x ) ≡ lim ⎜⎜
f 2 (t )dt ⎟⎟⎟
∫
⎜
Δx → 0 ⎜ Δx
⎟
⎝
⎠⎟
t =x
1/2
D (2)Q(x ) = ( DxQ(x ) )
26.5
Proof:
⎛ Q(x + dx ) − Q(x ) ⎞⎟1/2
D Q(x ) = Standard Part of ⎜⎜
⎟
⎜⎝
dx
⎠⎟
(2)
1/2
= ( DxQ(x ) )
85
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
27
Quadratic Mean Calculus
27.1 The Fundamental Theorem of the Quadratic Mean
Calculus
⎛ t =x
⎞⎟
(2) ⎜
(2)
Dx ⎜⎜ I f (t ) ⎟⎟⎟ = f (x )
⎜⎜ t =a
⎝
⎠⎟⎟
Proof:
⎛ t =x
⎜ (2)
(2) ⎜
Dx ⎜ I
⎜⎜ t =a
⎝
⎛ t =x
⎞⎟
⎞⎟
⎜
2
⎟
(2) ⎜
⎟
f (t ) ⎟⎟ = Dx ⎜ ∫ ( f (t ) ) dt ⎟⎟
⎟
⎜⎝⎜ t =a
⎠⎟⎟
⎠⎟
1
⎡ ⎛ t =x
⎞⎤
⎟⎟ ⎥ 2
2
⎢ ⎜⎜
= ⎢ Dx ⎜ ∫ ( f (t ) ) dt ⎟⎟ ⎥
⎟
⎢ ⎜⎜⎝ t =a
⎠⎟ ⎥⎦
⎣
= f (x ) .
27.2 Table of Quadratic Mean Derivatives and integrals
We list some Quadratic Mean Derivatives, and Integrals.
86
Gauge Institute Journal, Volume 4, No 4, November 2008
f (x )
1
2
a
D (2) f (x )
x
1
x2
2x
xa
ax a −1
x −1
ln x
0
0
0
I (2)f (x )
x
4x
a 2x
1 3
x
3
1 5
x
5
1
x 2a +1
2a + 1
−x −1
x −1
x
H. Vic Dannon
−1/2
ex
e x /2
e2x
2e x
eax
sin x
cos x
aeax /2
tan x
1
sin x
cos x
sin x
e sin x ( cos x ) e( sin x )/2
e cos x ( − sin x )1/2 e( cos x )/2
1/2
87
2
∫ ( ln x ) dx
1 2x
e
2
1 4x
e
4
1 2ax
e
2a
( sin2 x )dx
∫
∫ ( cos2 x )dx
∫ ( tan2 x )dx
∫ e 2 sin xdx
∫ e 2 cos xdx
Gauge Institute Journal, Volume 4, No 4, November 2008
H. Vic Dannon
References
[Abram]. Abramowitz, Milton and Stegun, Irene, “Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables”, United States Department of
Commerce, National Bureau of Standards, 1964.
[Bert] Bertrand, Joseph, “Traite de Calcul Differentiel et de Calcul Integral, Volume I,
Calcul Differentiel”, Gauthier-Villars, 1864. Reproduced by Editions Jacques Gabai,
2007.
[Doll]. Dollard, John, and Friedman Charles, “Product Integration with Applications to
Differential Equations”. Addison Wesley, 1979.
[Euler1] Euler, Leonhard, “Introduction to Analysis of the Infinite”, Book I, SpringerVerlag, 1988.
[Grob] Grobner, und Hofreiter, “Integraltafel”, Volume I, Springer Verlag, 1975.
[Haan] De Haan, D. Bierens, “Nouvelles Tables D’Integrales Definies”, Edition of 1867
– Corrected. Hafner Publishing.
[Kaza] Kazarinoff, Nicholas, Analytic Inequalities, Holt, Reinhart, and Winston, 1961.
[Melz] Melzak, Z., “Companion to Concrete Mathematics”, Wiley, 1973.
[Rain] Rainville, Earl, “Infinite Series”, Macmillan, 1967.
[Saks] Saks, Stanislaw, and Zygmond, Antoni, Analytic Functions, Third edition,
(Second is fine), Elsevier, 1971.
[Spieg] Spiegel, Murray, Mathematical Handbook of Formulas and Tables, McGraw-Hill
1968.
[Spiv] Spivey Michael, The Product Calculus, in ABSTRACTS of papers presented to
the American Mathematical Society, Volume 28, Number 1, Issue 147, p. 327.
[Zeid]. Zeidler, Eberhard, “Oxford User’s Guide to Mathematics”, Oxford University
Press, 2004.
88