Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 713241, 20 pages
doi:10.1155/2010/713241
Research Article
Functions on the Plane as Combinations of
Powers of Distances to Points
Jose L. Martinez-Morales
Institute of Mathematics, Autonomous National University of Mexico, A.P. 273, Admin. De corers No. 3,
62251 Cuernavaca, MOR, Mexico
Correspondence should be addressed to Jose L. Martinez-Morales, martinez@matcuer.unam.mx
Received 21 September 2010; Revised 26 November 2010; Accepted 22 December 2010
Academic Editor: A. Zafer
Copyright q 2010 Jose L. Martinez-Morales. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Some real functions on the plane can be expressed as a linear combination of powers of distances
to certain points.
1. Introduction
In this paper we construct a function basis to express different powers of the distance between
a point and the origin. Namely, for two given natural numbers m and n we find points in the
plane such that the function r n2m cosnθ can be expressed as linear combination of powers
of distances to those points see formula 3.15 below, both the linear combination and the
given function having the same domain in the plane for the notion of special function that
we use, see Section 3.1.
One of the important observations in the theory of real functions is that series
expansions lead to certain interesting numbers and functions e.g., Fourier coefficients, Lfunctions. Namely, usually series representations can be constructed explicitly in some
model spaces of sections of various function bundles over appropriate analytic manifolds.
By considering functions on the plane as maps F: R2 → R, the graphs of these
functions are the sections described by the map ψ: R2 → R3 ; x, y → x, y, Fx, y,
which defines 2-sub-manifolds on R3 . for analytic functions on the plane R2 ; see
Section 2.2.
We denote by AR2 the subspace of real analytic functions. It is well known that any
polynomial of degree n can be written exactly as a linear combination of powers of the right
number of terms of the form x · vk bk . We realize a series representation in the space of
analytic functions on the plane AR2 .
2
Abstract and Applied Analysis
Consider a point p in the plane R2 with coordinates r cos θ, r sin θ.
The distance used here is the Euclidean distance. Denote
|p| r.
1.1
We prove the following theorem.
Theorem 1.1. Consider two natural numbers m and n. There are N numbers a1 , . . . , aN and N
points p1 , . . . , pN , such that
r n2m cosnθ N
ai |p − pi |2mn .
1.2
i1
Remark 1.2. The numbers a1 , . . . , aN and the points p1 , . . . , pN can be determined exactly.
Corollary 1.3. The linear combinations of r n2m cosnθ, m, n 0, 1, . . ., can be expressed similarly.
Remark 1.4. An analogous theorem holds for the sine function.
Denote by x the maximum integer less than or equal to x.
Consider the function
δ x, y ⎧
⎨0 if x /
y,
1.3
⎩1 if x y.
2. Proof of Theorem 1.1
We will denote by pd the Legendre polynomial of degree d.
The following formula is formula 8.921 of 1:
√
1
1 − 2tx t2
∞
td pd x,
|t| < minx ± x2 − 1.
2.1
d0
√
Consider the set |t| < min |x ± x2 − 1| of R2 . Consider a subset H not necessarily proper of
such set. We will work solely with analytic functions on H.
i
ij i−j
Consider the series ∞
i0
j0 aij t x .
√
Divide this number by 1 − 2tx t2 . By formula 2.1, the following formula holds:
∞ i
i0
√
j0
aij tij xi−j
1 − 2tx t2
∞ ∞
i
aij tij xi−j · td pd x.
i0 j0
d0
2.2
Abstract and Applied Analysis
3
∞
We use the formula of the product of two series
n0
an and
∞
n0
bn :
∞
∞ n
∞
an bn ak bn−k .
n0
n0
2.3
n0 k0
The right side of formula 2.2 becomes
∞ i
aij tij xi−j ·
i0 j0
i−k
∞
∞ i td pd x ai−k,j pk xtij xi−k−j .
2.4
i0 k0 j0
d0
Let us change the order of addition with respect to the indices j and k:
i−k
i ai−k,j pk xtij xi−k−j i−j
i k0 j0
ai−k,j pk xtij xi−k−j .
2.5
j0 k0
The following formula is formula 8.911 of 1:
pk x k/2
2.6
bkl xk−2l ,
l0
where
bkl 1 −1l 2k − 2l!
.
2k l!k − l!k − 2l!
2.7
Substituting 2.6 in 2.5,
i−j
i ai−k,j pk xtij xi−k−j j0 k0
k/2
i−j
i ai−k,j bkl xk−2l tij xi−k−j
j0 k0
l0
k/2
i−j i 2.8
ai−k,j bkl tij xi−j−2l .
j0 k0 l0
Let us change the order of addition with respect to the indices k and l:
k/2
i−j ai−k,j bkl tij xi−j−2l i−j/2 i−j
k0 l0
l0
ai−k,j bkl tij xi−j−2l .
2.9
k2l
Denote by
aijl i−j
k2l
ai−k,j bkl .
2.10
4
The sum 2.9 is
Abstract and Applied Analysis
i−j/2
l0
aijl tij xi−j−2l . We will substitute this sum in 2.8.
i−j/2
k/2
i−j i i
ai−k,j bkl tij xi−j−2l aijl tij xi−j−2l .
j0 k0 l0
j0
2.11
l0
Let us change the order of addition with respect to the indices j and l:
i−j/2
i
j0
aijl tij xi−j−2l i/2 i−2l
aijl tij xi−j−2l .
2.12
l0 j0
l0
Let us add with respect to the index i from zero to infinity
i/2 i−2l
∞ aijl tij xi−j−2l .
2.13
i0 l0 j0
Let us change the order of addition with respect to the indices i and l:
i/2 i−2l
∞ aijl tij xi−j−2l i0 l0 j0
∞ i−2l
∞ aijl tij xi−j−2l .
2.14
l0 i2l j0
Change the index i for index i i − 2l:
i
∞ i−2l
∞ ∞ ∞ ij i−j−2l
aijl t x
ai 2l,j l ti 2lj xi −j .
l0 i2l j0
l0 i 0 j 0
2.15
We change the indices i and j for the indices
i i − l,
j j − l,
∞ ∞ i−l
∞ ∞ i
ai 2l,j l ti 2lj xi −j ail,j−l,l tij xi−j .
l0 i 0 j 0
l0 il
2.16
jl
Let us change the order of addition with respect to the indices i and l:
i ∞ i−l
i−l
∞ ∞ ail,j−l,l tij xi−j ail,j−l,l tij xi−j .
l0 il jl
i0 l0 jl
2.17
Abstract and Applied Analysis
5
Let us change the order of addition with respect to the indices j and l:
j
i−l
i i ail,j−l,l tij xi−j ail,j−l,l tij xi−j .
2.18
j0 l0
l0 jl
Denote by
Aij j
ail,j−l,l .
2.19
l0
The sum 2.18 becomes
i i
i−l
ail,j−l,l tij xi−j Aij tij xi−j .
2.20
j0
l0 jl
Let us add with respect to the index i from zero to infinity.
The ratio 2.2 becomes
∞ i
i0
√
j0
aij tij xi−j
1 − 2tx t2
i
∞ Aij tij xi−j .
2.21
i0 j0
2.1. Coefficients with Three Indices
Let us calculate the numbers ail,j−l,l . By formula 2.10, the following formula holds:
ail,j−l,l i2l−j
ail−k,j−l bkl .
2.22
k2l
Change the index k for index k − 2l:
i2l−j
ail−k,j−l bkl k2l
i−j
ai−l−k,j−l b2 lk,l .
2.23
k0
By formula 2.7, the following formula holds:
b2lk,l −1l 2l 2k!
.
22lk l!l k!k!
2.24
Let us use the formula
dn 1
−1n 2n!4−n x−n−1/2
√
dxn x
n!
2.25
6
Abstract and Applied Analysis
to write
−1k 2k xlk1/2 dlk 1
−1l 2l 2k!
√ .
l!k!
22lk l!l k!k!
dxlk x
2.26
We will substitute this in 2.23:
i−j
i−j
−1k 2k xlk1/2 dlk 1
ai−l−k,j−l b2lk,l ai−l−k,j−l
√ .
l!k!
dxlk x
k0
k0
2.27
Formula 2.22 becomes
ail,j−l,l i−j
ai−l−k,j−l
k0
−1k 2k xlk1/2 dlk 1
√ .
l!k!
dxlk x
2.28
2.2. A Series of Half-Integer Powers
Consider three numbers a, b, and c.
We will develop the power of the sum of three numbers.
Lemma 2.1. Consider a natural number n. Then, the following formula holds:
a b cn n i
i0
n!an−i bi−j cj
.
j0 n − i!j! i − j !
2.29
2
2 n
Consider the series ∞
n0 an a − 2abc c .
By Lemma 2.1 the following formula holds:
∞
∞
n i
n −1i−j 2i−j n!a2n−i−j bi−j cij
.
an a2 − 2abc c2 an
n − i!j! i − j !
n0
n0
i0 j0
2.30
We will distribute this sum, and we will exchange the order of addition:
∞ i ∞
n −1i−j 2i−j n!a2n−i−j bi−j cij
.
an a2 − 2abc c2 an
n − i!j! i − j !
n0
i0 j0 ni
∞
2.31
Denote by
aij ∞
an −1i−j 2i−j n!a2n−i−j
.
n − i!j! i − j !
ni
2.32
Abstract and Applied Analysis
7
The series becomes
∞
∞ i
n an a2 − 2abc c2 aij bi−j cij .
n0
2.33
i0 j0
Consider the formula
di xn a2n−2i n!
.
dxi xa2 n − i!
2.34
2.2.1. Formulas for Coefficients with Two Indices
Let us use formula 2.34 to rewrite formula 2.32.
∞
an −1i−j 2i−j ai−j di xn .
aij dxi xa2
j! i − j !
ni
2.35
Consider the function
fx ∞
an xn .
2.36
n0
Let us differentiate the series term by term:
∞
di xn
di
fx
a
.
n
dxi
dxi
n0
2.37
The i − 1 first terms in this series are equal to zero. Then, the following formula holds:
∞
di xn
di
fx
a
.
n
dxi
dxi
ni
2.38
−1i−j 2i−j ai−j di fx .
aij dxi xa2
j! i − j !
2.39
The number aij becomes
(1) Coefficients with Two Indices
We write the numbers ai−l−k,j−l substituting the numbers aij for numbers aij−1 aij .
By formula 2.39, the following formula holds:
ij−k−2l−1
a
ai−l−k,j−l
−1i−k−j 2i−k−j a2i−2k−2l−1 di−k−l fx .
dxi−k−l xa2
j −l ! i−k−j !
2.40
8
Abstract and Applied Analysis
Let us substitute in 2.28:
ail,j−l,l
i−j
−1i−k−j 2i−k−j a2i−2k−2l−1 di−k−l fx −1k 2k xlk1/2 dlk 1
√ .
l!k!
dxi−k−l xa2
dxlk x
j −l ! i−k−j !
k0
2.41
Let
2.42
x a2 .
Then, the following formula holds:
ail,j−l,l
di−k−l fx dlk 1 −1i−j 2i−j a2i
.
√ i−k−l dxlk x xa2
k0 j − l ! i − k − j !l!k! dx
xa2
i−j
2.43
The sum equals
ail,j−l,l
−1i−j 2i−j a2i di−j
i − j! j − l !l! dxi−j
dj−l fx dl 1
√
dxj−l dxl x
2.44
.
xa2
Let us add with respect of index l from zero to j:
j
l0
ail,j−l,l
j
−1i−j 2i−j a2i di−j
i−j
l0 i − j! j − l !l! dx
dj−l fx dl 1
√
dxj−l dxl x
xa2
j
dj−l fx dl 1 1
−1i−j 2i−j a2i di−j √ i − j!
dxi−j l0 j − l !l! dxj−l dxl x 2.45
.
xa2
The sum equals
j
l0
dj−l fx dl 1
1
1 dj fx
√
√ .
j − l !l! dxj−l dxl x j! dxj x
2.46
We will substitute this sum in 2.45:
j
−1i−j 2i−j a2i di−j 1 dj fx ail,j−l,l √ i − j!
dxi−j j! dxj x xa2
l0
−1i−j 2i−j a2i di fx .
√ i − j!j! dxi x 2
xa
2.47
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9
We will substitute this sum in 2.19:
−1i−j 2i−j a2i di fx Aij √ j!i − j! dxi x .
2.48
xa2
2.2.2. A Series of Half-Integer Powers
Divide both sides of 2.33 by
∞
√
a2 − 2abc c2 :
n ∞ i
i−j ij
an a2 − 2abc c2
i0
j0 aij b c
√
√
a2 − 2abc c2
a2 − 2abc c2
∞ i
i−j ij
i0
j0 aij b c
.
a 1 − 2bc/a c2 /a2
n0
2.49
By formula 2.21, the following formula holds:
∞ i
i−j ij
i
∞ i0
j0 aij b c
Aij cij bi−j ,
2
2
a 1 − 2bc/a c /a
i0 j0
2.50
where
i coefficients aij in 2.21 are replaced by coefficients aij−1 aij ,
ii coefficients Aij are given by formula 2.48.
The left side of 2.49 becomes
∞
n
i
∞ an a2 − 2abc c2
Aij cij bi−j .
√
a2 − 2abc c2
i0 j0
n0
2.51
2.3. A Series of Half-Integer Powers
n/2
Consider the series ∞
.
n0 an x
Let us separate this sum into
i a sum with respect to even indices,
ii a sum over odd indices.
Then, one has
∞
∞
∞
an x n/2 a2n xn a12n x1/2n
n0
n0
∞
n0
n0
a2n x n
∞
n0
a12n xn1
.
√
x
2.52
10
Abstract and Applied Analysis
In the second sum, we change the index n for index n n 1:
∞
∞
a12n xn1 a−12n xn .
2.53
n 1
n0
Define
a−1 0.
2.54
Then, the following formula holds:
∞
a−12n xn n 1
∞
a−12n xn .
2.55
n0
Formula 2.52 becomes
∞
∞
an xn/2 a2n xn n0
∞
n0
n0
a−12n xn
.
√
x
2.56
We will denote
fx ∞
a2n xn ,
n0
gx ∞
n0
2.57
a−12n xn
.
√
x
Let
2.58
x a2 − 2abc c2 .
By formulas 2.33 and 2.51, the following formula holds:
i
i
∞
∞ ∞ n/2 an a2 − 2abc c2
aij bi−j cij bij bi−j cij ,
n0
i0 j0
2.59
i0 j0
where see 2.39 and 2.48
−1i−j 2i−j ai−j di fx aij ,
dxi xa2
j! i − j !
−1i−j 2i−j a2i di gx bij j!i − j!
dxi .
xa2
2.60
Abstract and Applied Analysis
11
We will denote
−1i−j 2i−j ai−j
Aij j! i − j !
di fx dxi a
ij
xa2
di gx dxi .
2.61
xa2
Formula 2.59 becomes
∞
∞ i
n/2 an a2 − 2abc c2
Aij bi−j cij .
n0
2.62
i0 j0
2.3.1. The Case of the Law of Cosines
Consider a number θ.
Let
b cos θ.
2.63
The following formula appears in 2:
cos θn 21−n n!
n/2
cosn − 2kθ
.
k!n
− k!1 δn, 2k
k0
2.64
Let us use this formula to rewrite formula 2.62:
∞
an a − 2ac cos θ c
2
2
n/2
∞ i
n0
1−ij
Aij 2
i/2−j/2
i−j !
i0 j0
k0
cos i − j − 2k θ
cij .
k! i − j − k ! 1 δ i − j, 2k
2.65
By Lemma A.2 see the appendix, the following formula holds:
i
∞ 1−ij
Aij 2
i0 j0
i/2−j/2
i−j !
k0
j
i ∞ 21−ij A
i0 j0 k0
cos i − j − 2k θ
cij
k! i − j − k ! 1 δ i − j, 2k
i 2k − j ! cos i − j θ cij
ik,j−k 4
.
k! i k − j ! 1 δ i, j
−k
2.66
By formula 2.61, the following formula holds:
Aik,j−k
−1i−j 2i2k−j ai2k−j dik fx/dxik xa2 aij dik gx/dxik xa2
.
j − k ! i 2k − j !
2.67
12
Abstract and Applied Analysis
Multiply by 21−ij 4−k i 2k − j!/k!i k − j!1 δi, j:
21−ij Aik,j−k 4−k i 2k − j !
k! i k − j ! 1 δ i, j
−1i−j ai2k−j dik fx/dxik xa2 a2i2k dik gx/dxik xa2
2
.
j − k !k! i k − j ! 1 δ i, j
2.68
Let us use formula 2.34 to rewrite
−1i−j ai2k−j dik fx/dxik xa2 a2i2k dik gx/dxik xa2
2
j − k !k! i k − j ! 1 δ i, j
−1i−j dj−k xi /dxj−k 2 a−ij dik fx/dxik xa2 a2j dik gx/dxik xa2
xa
.
2
j − k !k! 1 δ i, j i!
2.69
Let us add with respect to the index k from zero to j:
j
−1i−j dj−k xi /dxj−k 2 a−ij dik fx/dxik xa2 a2j dik gx/dxik xa2
xa
2
j − k !k! 1 δ i, j i!
k0
−1i−j a−ij dj /dxj xi di fx/dxi xa2 a2j dj /dxj xi di gx/dxi xa2
.
2
i!j! 1 δ i, j
2.70
Multiply by cosi − jθcij .
Let us add with respect to the index j from zero to i.
Let us add with respect to the index i from zero to infinity.
By Lemma A.1 see the appendix, the following formula holds:
−1i−j a2j dj /dxj a−i−j xi di fx/dxi xi di gx/dxi 2
xa
cos i − j θ cij
2
i!j! 1 δ i, j
j0
i
∞ i0
∞ ∞
cos jθ c2ij
−1j di /dxi xij dij /dxij a−j fx a2i gx
.
2
i! i j ! 1 δ 0, j
j0 i0
2.71
We will denote
−1j di /dxi xij dij /dxij a−j fx a2i gx 2
xa
Bij 2
.
i! i j ! 1 δ 0, j
2.72
Abstract and Applied Analysis
13
The left side of 2.65 becomes
∞
∞
∞ n/2 an a2 − 2ac cos θ c2
Bij cos jθ c2ij .
n0
2.73
j0 i0
(1) A Formula for Calculating Coefficients
Consider two natural numbers m and n.
Let
fx xmn ,
gx 0.
2.74
Formula 2.72 becomes
Bij 2
−1j a−j2 m2n−2i m n!2
.
m n − i! m n − i − j !i! i j ! 1 δ 0, j
2.75
3. Functions on the Plane Expressed as a Linear Combination of
Powers of Distances to Points
We deduce an identity regarding linear combinations of powers of distances from certain
points, as well as similar new identities, from the formula of a power of the cosine. The
formula of a power of the cosine is a powerful tool in mathematical analysis. In the space of
p
analytic functions, we construct sums N
i1 ai |p − pi | associated to them. We allow ourselves
to use the name sum even if it is not finite.
Consider
i two numbers r and θ,
ii two natural numbers m and n.
Assume that there are i numbers θ1 , θ2 , . . ., ii positive numbers p1 , p2 , . . ., and iii series
∞
2
2 i/2
, k 1, 2, . . ., such that
i0 aki pk − 2pk r cosθ − θk r r n2m cosnθ ∞
i/2
aki pk2 − 2pk r cosθ − θk r 2
.
k
3.1
i0
We will denote
fk x ∞
ak,2i xi ,
i0
gk x ∞
i0
ak,−12i xi
.
√
x
3.2
14
Abstract and Applied Analysis
By formula 2.73, the following formula holds:
∞
∞ ∞
i/2 aki pk2 − 2pk r cosθ − θk r 2
Bkij cos jθ − θk r 2ij ,
i0
3.3
j0 i0
where the coefficients Bkij are given by formula 2.72:
Bijk
−1j di /dxi xij dij /dxij a−j fk x a2i gk x 2
xa
2
.
i! i j ! 1 δ 0, j
3.4
We use the formula
cos θ − φ cosθ cos φ sinθ sin φ .
3.5
Formula 3.1 becomes
r n2m cosnθ ∞ ∞
Bkij cos jθ cos jθk sin jθ sin jθk r 2ij .
3.6
j0 i0
k
The following formulas are sufficient conditions for the solution of the equation above:
kπ
, k 1, 2, . . . ,
n
jkπ
Bkij δi, mδ j, n ,
cos
n
k
θk k
3.7
jkπ
sin
Bkij 0,
n
i, j 0, 1, . . . .
3.1. The Product of a Power of the Distance to the Origin and Cosine
Consider
i 1 m numbers a0 , . . . , am
ii 1 m positive numbers p0 , . . . , pm .
We will denote
fkl x −1k al xmn ,
k 1, . . . , 2n; l 0, . . . , m.
3.8
In Section 2.3.11, we computed the numbers Bij . Let us denote
Bkl,ij 2
−1kj al pl −j2m2n−2i m n!2
.
m n − i! m n − i − j !i! i j ! 1 δ 0, j
3.9
Abstract and Applied Analysis
15
Multiply by cosjkπ/n sinjkπ/n, resp., and add up over the index k from one to 2n,
2n
2n
k1 Bkl,ij cosjkπ/n l1 Bkl,ij sinjkπ/n, resp..
We use the formula
2n
√
−1k e −1jkπ/n 2nδ j, n .
3.10
k1
The sums
2n
k1
Bkl,ij cosjkπ/n and
2n
jkπ
Bkl,ij cos
n
k1
2n
k1
Bkl,ij sinjkπ/n are equal to
4nδ j, n −1j al pl −j2m2n−2i m n!2
m n − i! m n − i − j !i! i j ! 1 δ 0, j
4nδ j, n −1n al pl n2m−2i m n!2
,
m n − i!m − i!i!i n!1 δ0, n
3.11
2n
jkπ
0.
Bkl,ij sin
n
k1
We want the numbers al to satisfy the following equation:
2n
m l0
jkπ
Bkl,ij cos
n
k1
δi, mδ j, n ,
i 0, . . . , m; j 0, 1, . . . .
3.12
By formula 3.11, this sum is reduced to
m
δi, m1 δ0, n−1n n − 1!m!
,
al pl n2m−2i 4m n!
l0
i 0, . . . , m.
3.13
This is a system of linear equations with unknowns a0 , . . . , am . The coefficient matrix for the
system is pl n2m−2i ; i, l 0, . . . , m.
Assume that the numbers p0 , . . . , pm are different from each other. Then,
al 1 δ0, n−1n n − 1!m! −n pi2
pl
,
2
2
4m n!
i/
l pi − pl
l 0, . . . , m.
3.14
Equations 3.7 are valid. Therefore,
r n2m cosnθ 2n
m mn
,
−1l al pl2 − 2pl r cosθ − θk r 2
3.15
l0 k1
where, as in 3.7,
θk kπ
,
n
k 1, 2 . . . .
3.16
16
Abstract and Applied Analysis
2π/n
π/n
2π
Some angles θk , n 5
Figure 1
2π/n
π/n
pk−1
pk
2π
Points with coordinates pl cos θk , pl sin θk , k 1, . . . , 2n; l 0, . . . , m
End of proof
Figure 2
3.2. Example: The Difference between Two Paraboloids
Write
r cosθ r cosθ
1
r2
2
16
1
r cosθ
r2 .
− −
2
16
3.17
In the coordinates x, y,
r cosθ x.
Figure 3 represents graphically Identity 3.17.
3.18
Abstract and Applied Analysis
1.2
1.2
0.8
0.8
0
−0.4 −0.2
−1 −0.8 −0.6
0
17
x
0
−0.4 −0.2 0
−0.5
−1
0.2 0.4
−0.5
−1
0.2 0.4
0.6 0.8
x
1
1
0.5
1
−1
y 0.5
0
−0.5 −0.5
−1
−0.5
0.5
x
1
−1
Figure 3
3.3. Example: A Combination of Fourth-Degree Surfaces
Write
r 2 cos2θ −
1
1
− √ r sinθ r 2
8
2
−
2
1
1
√ r sinθ r 2
8
2
2
1
1
√ r cosθ r 2
8
2
2
1
1
− √ r cosθ r 2
8
2
3.19
2
.
In the coordinates x, y,
r 2 cos2θ x2 − y2 .
Figure 4 represents graphically Identity 3.19.
3.4. Example: The Product of a Bessel Function and Cosine
Consider the product r n
∞
m0 −1
m
/m!n m!r 2m cos nθ.
3.20
18
Abstract and Applied Analysis
−0.5
x
0.5
4
4
3
3
2
2
1
1
0
0
0.5
x
−0.5
−1
1
1
y
−1
4
4
3
3
2
2
1
1
0.5
0
−1
−0.5
x
0.5
y
0
−0.5
−1
1
1
1
0.5
y
0.5
0
−0.5
−1
−1
Figure 4
p − pkl
p
pkl
θ
θl
Figure 5: Representation of the difference p − pkl .
x
1
Abstract and Applied Analysis
19
0.4
2
0.4
3
0
3
2
0.2
0.2
0
0
−1
−1
−2
−3
−2
−3
−3
3
−3
0.4
0.4
0.2
0.2
3
0
−3
−1
0
−1
−3
−2
−2
−3
−3
Figure 6: Graphics of both sides of formula 3.22, n 2, 3. In the series on the right, we add up to m 6.
We will denote
aklm pi 2
1
.
−1nkm 1 δ0, npl −n n!m n!−2
4
p 2 − pl 2
i/
l i
3.21
By formula 3.15,
rn
2n ∞ m
mn
−1m
r 2m cos nθ aklm pl2 − 2pl r cosθ − θk r 2
.
m!n m!
m0
k1 m0 l0
∞
3.22
We will denote
p : point with coordinates r cos θ, r sin θ,
pkl : point with coordinates pl cos θk , pl sin θk ,
3.23
|p| r, y
skl x xn
∞
ml
aklm xm .
20
Abstract and Applied Analysis
Rewrite formula 3.22 as
rn
∞
2n −1m
r 2m cos nθ skl |p − pkl |2 .
m!n m!
m0
k1 l0
∞
3.24
Appendix
A. Two Lemmas
We give two formulas for changing the order of the sums in double series.
i
Lemma A.1. Consider the series ∞
i0
j0 ai,i−j . Assume that one can exchange the order of addition
with respect to the indices i and j. Then, the following formula holds:
∞ ∞ i
∞
aij aij,i .
i0 j0
A.1
j0 i0
Lemma A.2. Consider the series
i/2 i−2k
i ∞
i0
j0 aijk y,
k0
∞ ∞ i
ii k0 ik jk aik,j−k,k .
Assume that one can exchange the order of addition with respect to the indices i and k. Then, the
following formula holds:
i−j/2
i
∞ i0 j0
k0
aijk j
i ∞ aik,j−k,k .
A.2
i0 j0 k0
References
1 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, San Diego,
Calif, USA, 6th edition, 2000, Translated from the Russian, Translation edited and with a preface by A.
Jeffrey and D. Zwillinge.
2 http://en.wikipedia.org/wiki/List of trigonometric identities.