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PROPERTIES
"ABLES
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OF THE EXTENDED
AIRY-HARDY INTEGRALS
M. V. CERRILLO
W. H. KAUTZ
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TECHNICAL REPORT 144
NOVEMBER 15, 1951
RESEARCH LABORATORY OF ELECTRONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
CAMBRIDGE, MASSACHUSETTS
/VL
~
The Research Laboratory of Electronics is an interdepartmental laboratory of the Department of Electrical Engineering
and the Department of Physics.
The research reported in this document was made possible
in part by support extended the Massachusetts Institute of Technology, Research Laboratory of Electronics, jointly by the U. S.
Army (Signal Corps), the U. S. Navy (Office of Naval Research),
and the U. S. Air Force (Office of Scientific Research, Air Research and Development Command), under Signal Corps Contract DA36-039 SC-64637, Project 102B; Department of the
Army Projects 3-99-10-022 and DA3-99-10-000.
MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
RESEARCH LABORATORY OF ELECTRONICS
Technical Report 144
November 15, 1951
PROPERTIES AND TABLES OF THE
EXTENDED AIRY-HARDY INTEGRALS
M. V. Cerrillo
W. H. Kautz
Abstract and Foreword
1.
Airy-Hardy and other associated functions play an important role in the
theory of approximate integration; specifically, in saddlepoint methods. They appear
in the pure and mixed transitional cases corresponding to configurations that lead to
saddlepoints of third order.
(For definitions and other details see references 1 and 2.)
There are three basic Airy-Hardy functions.
Ah 3 3.
They are denoted by: Ah 1 3, Ah2 3
The first index refers to the particular function; the second index expresses
the order of the saddlepoint associated with the function.
sentations, and properties are given in this report.
Definitions, series repre-
The connections of these functions
with the methods of integration are supplied in Appendices I and II.
2.
The functions Ah V 3, v = '1, 2, 3, correspond to the so-called pure transitions.
The function Ah 1 , 3(s) has been tabulated for complex values of the argument s.
function represents a surface over the s-plane.
This
Radial cross sections have been com-
puted for angular arguments in steps from 0 ° to 180 ° and for Isl in the range from
zero to 4. Twenty-seven tables give Ah 1, 3 (s), arg {Ah 1 3(s)}, Re{Ah 1, 3(s)}, and
Im {Ah 1 , 3(s)}. The plot of Ahl, 3 (s) is expressed in the graphs of the cross sections
of each of the functions above. Several isometric plots of the corresponding surfaces
are also given.
3.
The aim of the computations given in this report is to show an over-all
behavior of the functions indicated in paragraph 2 above, rather than to produce a
detailed evaluation of those functions.
The computations shown in these tables were
made on small mechanical machines during the summer of 1950.
4.
Only the function Ahl, 3 was tabulated.
between this function and Ah 2 3 and Ah3 3
There are simple connecting relations
that permit a rapid computation for Ah2, 3
and Ah 3 , 3 from the tables of Ah 1 , 3'
5.
0V
Mixed transitions of third order require some other functions,
v,p(B) and
o(B), which appear when the saddlepoint of third order runs over or in the vicinity
of a pole or a zero of the integrand.
(See Appendices I and II. ) These functions can
be expressed as the integrals or derivatives, respectively, of the Ah 1 3 functions.
The functions,
P v,
p(B) and
v ,o(B), are not tabulated in this report; however, the
corresponding series representations are given.
A simple numerical integration or
differentiation can be performed from Ahl, 3, and the evaluation of q 1, p (B) and P 1, o(B)
obtained without much effort. The figures of transient composition given in Appendix II
were obtained with ease by this method.
6.
This report was originally planned as a part of the Series 55 reports.
Later,
its publication as a separate report seemed appropriate because the material, which
is self-contained, is applicable in other fields and may prove more useful if it is
published separately.
The authors want to express their thanks to Mrs. Ann Moldauer for her help
in preparing the numerical computations involved in this report; to Miss Elizabeth
7.
Campbell for her constant advice and suggestions; and to other members of the Joint
Computing Group for their careful proofreading of the tables.
Manuel V. Cerrillo
William H. Kautz
May 10, 1956.
ii
Table of Contents
Abstract and Foreword
i
I.
Properties of Extended Airy-Hardy Integrals
I. 1 Definitions of the Functions and Their Interrelations
1
I. 2 Power Series Expansion
2
I. 3 Relation of Ah,
2
1
3 to Other Functions
I. 4 Differential Equation Satisfying Ahv 3
4
I. 5 Asymptotic Expansions for Large IBI
4
II.
Tables of the Extended Airy-Hardy Integrals
9
II. 1 Preparation of the Tables
9
References
11
Tables 1-25.
Numerical Computations of Re Ah1 , 3 (B), Im Ah,
Ah1 , 3 (B), Arg Ah 1
Im Ah3, 3 (B)
3
(B), Re Ah 2 , 3 (B), Im Ah2,
3
(B), Re Ah 3
3
3 (B),
(B),
12
k
Table 26.
Coefficients of B
Table 27.
Zeros of Ah 1 , 3 (B) for B Real
in the Series (I. 2)
37
38
Plots of Extended Airy-Hardy Integral
39
Appendix I
53
Appendix II
61
iii
I.
PROPERTIES OF EXTENDED AIRY-HARDY INTEGRALS
I. 1 DEFINITIONS OF THE FUNCTIONS AND THEIR INTERRELATIONS
In certain branches of applied mathematics - in a new theory of the transient analysis
of linear electrical systems (1 ) , in particular - there arise a number of important complex
prototype integrals, the evaluation of which can be accomplished only in terms of littleknown functions.
One set of these integrals, the Lommel generating functions, was
mentioned in a previous report (2 ) , and discussed and tabulated in greater detail in
another report ( 3 ) .
A second set of integrals of basic importance is:
Ah,, 3 (B) =
,f
eBz+z3 dz
;
1(I. 1)
v = 1,2, and 3
The contours y 1 , 2 , and y3 corresponding to the three functions Ahl,
and Ah3, 3 (B) are those shown in Fig. la, b, and c. It is apparent that
Ah2, 3 (B),
3 (B),
2(I. 1)
Ahl, 3(B) = Ah2,3(B) + Ah3,3 (B)
Since the integrand is an entire
function, these contours may be
distorted from the positions shown
as long as they pass to infinity
in directions that do not deviate
more than 30
of Fig. 1.
°
z- PLANE
z-PLANE
z- PLANE
r2<
r3L
from the directions
Under these con-
ditions, the integrals converge
(b)
for all finite B, real or complex,
and therefore define three
(c)
Contours pertinent to the evaluation of
Fig. 1.
extended Airy-Hardy integrals.
single-valued, entire functions.
To establish other relations
between the three functions, let
r
/3 in the integrals for A.h 2 , 3 (B) and Ah3, 3 (B).
z-ze±i27
Ah2 3 (B) } =
Then
eBe±i2/ 3z+z3 2dz . ei
2
7/3
or
= _ e+i2t3 . Ah 3 (Be+i27/ 3 )
Ah2,3 (B)
Ah3,3 (B) J
The three functions are, therefore, not independent of one another.
one determines the other two completely.
1
3(I. 1)
The knowledge of
I. 2 POWER SERIES EXPANSION
For purposes of computation, it is helpful to have the power series expansion of,
let us say, Ah 1
3
(B).
The uniform convergence of the development
eBz
(Bz)k
E
k!
k=O
guarantees that if this series is substituted in the first integral of 1(I. 1) the processes
of integration and summation can be interchanged.
Ahl, 3 (B) =
Bk
1
The resultant sum
kz 3
z
k= k 2
e
dz
can be evaluated term by term. If we let z = re- / 3 on the lower part of the contour
(Fig. la), and z = re i 7T/ 3 on the upper part, we obtain
Ahl, 3(B)=
ke
-1 {e-i(k+1)7e/3
k=O k! 2Ri
E
Bk
0
f
+ ei(k+l),/3
rkedr
00
f
rke-r3
dr
Hence,
Ah,
3 (B)
E
Ah1 (B)=
{6-
-1, 0, ...
3
3
1)
sin [(k
This is the desired power series.
-1,
Bk sin [(k+ 1)r]f rker
dr
Bk
1(I. 2)
The quantity in braces takes on the values 1, 1, 0,
cyclically as k = 0, 1, 2, ....
At B = 0,
Ahl, 3 (0) =
F(1/3) = 0.246162704
6?
The radius of convergence of 1(I. 2) is
R =
im
k-*-0
r(k+1/k!Il/k
k 3
II
as it should be for an entire function; and the order p is given by
1im
log(kg!/r(+
1/P
))
3
k--,
k log (k)
2
2
3
Hence, p= 3/2.
I. 3 RELATION OF Ahv, 3 TO OTHER FUNCTIONS
Certain connections between the extended Airy-Hardy integrals and other functions
can be derived by manipulation of the terms of the power series l(I. 2). For example,
to express Ahl, 3 (B) in terms of Bessel functions, make the change of index
2
(k + 1)/3 = m + 1 in 1(I. 2) and write the terms in the form:
Ahl, 3 (B)
63 E
s in (m
[
+ 1)r]
I-
m-O~
r'(m+ 1) B3m+2 +
1 (3m + 2)!
[
21
31T]
sin (m
sin (m
+
+ 1 7
3/ I
(3Im)!
m+]
\(
(3m)!
m
The first bracket is zero; the second and third are each equal to (-1)
'(m + )B3m+1
(- 1)m
H
Ah, 3(B) =
.
3
B
B3m
Hence,
F(rm +)B3W
+
mO
61
2-)B3m+1
(m
(3mr + 1)!
3/
(3m)!
(3m + 1)!
1(I. 3)
)
By making use of the identity,
Frm+_) r(m+ 2 )= 2r
3} V
3J \
.
(3m)!
33mm!
the gamma functions in the numerators of 1(I. 3) can be shifted to the denominator,
giving
E
Ahl,3 (B)
(_ l)m
3m=O m! 33m
B3m+l
B3m
+
j
P(m +3)(3m+ 1)
or
00
Ah1,3 (B) =
(-) m
m=O m!
E
{
(B/3)3m+l
(B/3)3m
+3
)
r (m +4)
[(B)3/2] 2m+
Ahl, 3 (B)
1
3
(
(B)1/2
1)m
+
m=O
B3/2] 2m-1
r(m + 2
r (m+ 4 )
These two sums can now be identified with those for the Bessel functions of the first
kind of orders 1/3 and -1/3:
Ahl,3 (B) =
j1/3(2[]3/2)
3}
+ J 1/ 3 (2[B 3/2}
2(I. 3)
If the development of 1(I. 2) is carried out in terms of -B instead of B, we obtain:
(B)
Ah
1,3(B)
1
1/(
[B]3/2)
'I[=B3/2)
3
1/3(
3
K1 /3 ([
B] 3 / 2 )
3(I. 3)
Since the expansions of J 1/ 3 (z) and I 1 / 3 (z) converge for all finite z exterior to a cut,
which may be placed along the negative real z-axis, the expansions 2(I. 3) and 3(I. 3)
3
converge for arg(B)J < 2 r/3 and I arg(-B) I < 2zr/3, respectively. Hence, the entire
finite B-plane is covered, and the functions Ah 2, 3 (B) and Ah3, 3 (B) can be similarly
expressed in terms of Bessel functions through the relation 3(I. 1).
Known relations between the Bessel functions and the confluent hypergeometric
function lead to the representation:
Ah,
3 (B)
= Bei2(B/2) 3/ 2
3r (1/3)
1F(
i4[B] 3 /
5 ;
1
)
3
+ e- i2(B/2)
/23
31 ;
3r (2/3)
T
2;
4(.3)
Ahl, 3 (B), Ah2, 3 (B), and Ah 3 3 (B) are actually generalizations of combinations of
certain functions of Airy, Ai(x) and Bi(x), defined for real argument
The Airy
integral is defined by
1
Ai(x)=
Ri(x)
e
x t + ( 1/ 3 ) t 3
dt
so that the equivalence
Ahl, 3 (B)
follows.
3-1/3 Ai (- 3-1/3 B)
5(I. 3)
In a similar fashion,
Ah2,3(B) }
Ah3, 3 (B)
3-_1/3 [Ai(- 3-1/3B)
iBi (- 31/3 B)]6(L
3)
2
where Bi(x) is the Airy integral of the second kind. Both Ai(x) and Bi(x) [also Ai'(x)]
are well tabulated for both positive and negative real values of x, and their zeros,
turning points, maxima, and minima are known over a wide range ( 4 )
I. 4 DIFFERENTIAL EQUATION SATISFYING Ahv 3
Double differentiation of the series 1(I. 2) yields a series of almost the same form
as 1(I. 2) itself. Closer inspection reveals that
Ah,
3 (B)
= -
3
Ahl, 3 (B)
1(I. 4)
In terms of the Airy integral, this differential equation reads
Ai"(x) = xAi(x)
2(1. 4)
I. 5 ASYMPTOTIC EXPANSIONS FOR LARGE I B
The most direct method for obtaining an asymptotic series to represent Ahl 3 (B)
[hence, Ah2, 3 (B) and Ah3, 3 (B)] from 3(I. 1) for large I B I is that of utilizing the known
expansions for the Bessel functions in the relations 2(I. 3) and 3(I. 3). For
Iarg(y) <n/2, Watson ( 5 ) gives:
4
J±1/3 (Y -
F(
Vr--wfY-Cos Y
6
- sin (y
\
6
m4
-O
+ 2) - - I)
4 m-
1 - 2m) (2m)!
r(l
¥ i 3
r(i
r(
1 + 2m+ l)
(-)
I
(2y)2.
m
1
(2y)2m+l
- 2m - 1)(2m + 1)!
which can be written in the more abbreviated form:
2
Iry
J1/3 (Y)-
1
1 + >
(
2
1 - m) m!( 2y)m sin (y
3
( 1+m)
2
m0
m=0
1
+ mr
+
+- 7
6
4
2
1(I. 5)
and
K1/3 () -e
For large [ B I , then, and for
(5 + m)
E
1
6
mO r(F - m) m! (2y)m
6
2Y
2(I. 5)
I arg(B) I < ir/3, relation 2(I. 3) yields
+m
r5
Ah1,3 (B) - 1
m![4(B/3) 3/ 2] m
r(5 - m)
sin (y
-_
+
\Y
4
6
F(l-+ m)
1+ 1(
-
m
sin (Y
+ 7
+
4
)
7 + m)I
6
2)
or, better,
Ah,3 (B )
For I arg (-B)
I _V773rB
VM
(- )m (6m)!
sin (y + (2m + 1) )
(2m)! (3m)! (4\I3 )3m
4
3(1. 5)
< i7/3, relation 3(I. 3) yields
Ah, 3 (B)
e'2(-B/3) 3/2
(6 + m)
0
2 W\V
1
5 - m) m! [4 (-B/3)3/2 ] m
m=O
or
AhI, 3 (B)
3
e2(B/3)/2
2X/
f
- m=o
(- l)m (6m)!
(2m)! (3m)! (4i'-3B)3m
4(I. 5)
5
-
-----
-
An alternative expansion, more suited than 3(I. 5) to computation when B is not real,
can be derived by applying the saddlepoint method of integration to the defining integral
for Ahl, 3 (B), namely,
Ahl, 3 (B) =
J
1
eBZ+Z 3 dz
5(. 5)
This method is based upon the fact that the magnitude of the integrand of this integral,
when considered as a function of z, is relatively small over all of the contour y 1 except
that portion near z = 0, where this magnitude assumes larger values. Consequently,
almost all of the value of the integral could be obtained by integration over only this
short section of contour where the integrand is large. The approximation to the true
value of the integral should improve as the integrand becomes smaller along the tails
of contour.
The process of integration along the contour y1 can be visualized as a passage over
mountainous terrain that represents the magnitude of the integrand, which, for
i
z = re
, is
eReIW(z)} _ erb cos (+,8) + r3 cos (30)
where B = bei
/.
For large r, it is clear that there are three ridges of increasing
altitude in the directions 0 ° , +120 ° , and -120 ° , separated by three valleys of increasing
depth in the directions -60 ° , 60 ° , and 180 ° . Near r = 0, these three ridges and three
valleys conflow to form two mountain passes, or "saddlepoints, " at which points the
terrain is flat. These points are given by
dW
0
B + 3z2
-3
- + +e
dz
or
z z
i ( T+ )
6(I. 5)
In order to carry out the integration along the contour near z = 0, we must distort
the contour from the position shown in Fig. la in such a way that the resultant line
integral can be evaluated in terms of known functions. It would not be illogical to try,
first, a pair of connected contours that pass directly through the lower and upper
saddlepoints, oriented in a convenient direction. With this choice of contour in mind,
let the exponent W be expanded in a Taylor series about either saddlepoint Zs:
W(zs) + W(zs)(-Zs) +
W(z)
2
W"(Z)(z
zs)2 + . .
or
W(z)
=2
(- B/3) 3 / 2 ±
3B (z
-z)2
6
+ (z -z)3
7(. 5)
7-B/3,
and the lower sign to
where the upper sign refers to the saddlepoint at z = +
i
-B/3. We now let z - z s pe' , and choose a convenient
the saddlepoint at z = for each section of the contour. With this substitution in 6(1. 5), the integral
value of
for Ahl, 3 (B) becomes
Ah1 3 (B) =
1 e2(-B/3)3/2fPl e+/3B p2ei2i ep3i31 dpe il
X7i
-~
2
1
e-2(-B/3) 3/
2
2fi
2e
p3
2
32 d2ei2eI2
8(I. 5)
-P2
These two integrals represent the contributions along the lower and upper contours,
respectively.
The angles
1
and 02 are the inclinations of the contours, and the distances
P 1 and P 2 are measured from the corresponding saddlepoints to the point of intersection
of the contours. This arrangement of contours is shown in Fig. 2 for 3 = arg(B) = 40 ° .
These integrals would be simplified
for evaluation if the angles 01 and 02
could be chosen to make the exponential
terms involving p negative real. This
choice of contour directions corresponds
to the directions of steepest descent from
the "mountain passes" at the saddlepoints
to the valleys below, since any other
choice of 01 and 02 would make the real
multipliers of p 2 smaller, thus indicating
a more gradual decrease in altitude on the
two sides of each pass. As the contours
pass into the valley, they deviate from the
paths of steepest descent (shown as dotted
lines in Fig. 2), because of the effect of
the p3 term. For large I BI , the size
of the integrand is controlled mainly by
Fig. 2. L-shaped contour pertinent to the
evaluation of the first extended AiryHardy integral by the saddlepoint method.
the second-degree term, so that the
departure of the contours from the bottom of the valleys is not great over significant
sections of the contours.
The p 2-coefficients are apparently negative real, provided that
_7 +
2
2
+22 1 =
or
1 = 3r
4
_
4
and
9(I. 5)
+
2 -2
+ 22 =
,
or
4
)2
4
7
-
-
The evaluation of the integral is now a straightforward process. The p3-terms,
being small in comparison with the p 2-terms for large I B I, can be removed from the
exponent with the rapidly convergent development,
eP 3 e 3'o
p3
m=O
ei3m-
m!
The integrals 6(I. 5), therefore, become
Ah 1,3(B)
j
eei2(B/3)3/2+i1
.
m
2w
inom
Id0
2rri
2ffi
p3me-P
ei3mO2
2·imz
2
3m-p
!Or
+ei2(B/3) 3 / 2 +i 2
e/
+
Pl
e3ml
i::)mh
If
mO
m!
,p3 e
2
2 dp
p dp
0(I. 5)
10(.
5)
P2
After a considerable amount of algebraic computation, an asymptotic formula for
Ahl, 3 (B) for large I B and for fi < 60 ° is obtained:
Ah 1 (B)
2tf mOm! (3B)( 3m+l)/ 4
s
sin(
(3+ 1
y+ (3m+1)
1(I. 5)
3m
1((( 5)
Iyl[-sin(/2)]d
)
12(I.5)
+
4Pm(B)
)_R)
where y = 2(B/3)3/2, and
Pm(B) =
Qm(B)
(3m+1)
=
()m
2
+ (-1)m {r(3m+ 1
r(3m+1 ;
\
l
sin/2)])+ r( 3m+)}
yj [1l+sin((8/2)]
2
-
(3m+
i
The latter functions are incomplete gamma functions, defined by
r(x;p)
fP txle-tdt
0
and are well tabulated( 6) .
The formula 11(I. 5) is actually a refined form of the asymptotic development
3(1. 5) derived from Bessel-function expansions. Equation 3(1. 5) can be obtained from
11(I. 5) by extending the tails of the two sections of the L-shaped contour of Fig. 2 to
infinity - that is, by letting P 1 and P 2 approach infinity so that both of the integrals in
10(I. 5) have doubly infinite limits. Since the contributions to the integrals along these
added portions of the contours are small, especially for large IB , we would expect
a significant improvement in the accuracy of the asymptotic series only for moderate
values of I B I . When P 1 -
-
oo and P 2 -- o in 10(I. 5), then the incomplete gamma
function becomes the complete gamma function in 11(I. 5):
lim r(x;p)
p-~OO
8
r(x)
The equivalence of 11(I. 5) and 3(I. 5) now follows from elementary properties of the
gamma functions.
When = 180 ° (B negative real) the two saddlepoints lie on the real z-axis. A
single contour through the right-hand saddlepoint is equivalent to the original y 1 -contour
of Fig. la. (The right-hand saddlepoint is chosen because the directions of steepest
descent are oriented normal to the real axis for this saddlepoint, but lie along the real
= 90'; hence, z- z s = ip, and
axis for the left-hand saddlepoint. ) For this contour,
the integral corresponding to 8(I. 5) becomes
l
Ah1,3(B)
2,ffr
e 2(-B /3) 3/ 2 f
eV'p
2
'ip 3 dp
13. (I. 5)
The integration is straightforward, and yields the result 4(I. 5) that would be expected
from the relationship of 11(I. 5) and 3(I. 5).
II. TABLES OF THE EXTENDED AIRY-HARDY INTEGRALS
II. 1 PREPARATION OF THE TABLES
The tables at the end of this report give the real and imaginary parts, magnitude,
and angle (in radians) of the first extended Airy-Hardy integral, Ah 1 3 (B) = Ahl 3( B ei/),
and the real and imaginary parts of the second and third functions, Ah2, 3 (B) and
Ah 3 , 3 (B). The magnitude and angle of Ah2, 3 (B) and Ah 3 3 (B) can be read from the
appropriate tables for the magnitude and angle of Ahl, 3 (B), using the relation 3(1. 1),
which can be written:
Ah1,3(Bei120)1
lAh2 ,3 (B)I
Ah,
Ah 3, 3 (B) I
3 (Be
il
2
rn
1(n. 1)
Ah 2 ,3 (B)
g{ Ah ,3 (B)
j
rg
Ah13 (B) T 600
All values are given to seven decimal places (last figure uncertain, particularly in the
values of arg Ahl, 3 (B)) for the range of the independent variable,
IBI
0(0.2)4
, = 0(7.5)180°
;
Ahl, 3 (B) is real for Re {BI; therefore, for negative values of
Ah1,3 (B) - Ahl, 3 (B)
, we have
2(II. 1)
The values in the tables were computed by direct calculation of ReAhl, 3 (B) and
ImAhl, 3 (B) from the series 1(I. 2). The second and third columns in each of
Tables 1-25 were computed directly from the first two columns; the last four columns
were computed from the first two by use of relation 3(1. 1).
9
Table 26 gives the coefficients of the series 1(I. 2) to eight significant figures.
Table 27 gives the zeros of Ah 1
3 (B)
for Re {B}. This table was taken from ref-
erence 4 (page 343), with the appropriate change of scale.
Figures 3-5, which follow
the tables, are plots of the functions computed along rays p = constant.
The figures
are self-explanatory.
The natural extension of these tables to larger values of B is through the asymptotic
expansions 3(I. 5) or 11(I. 5) and 4(I. 5).
Existing tables of the Airy-Hardy integral in various forms are listed in references 4, 5, and 7 for real argument; in references 8 and 9 for complex argument.
10
References
1.
M. V. Cerrillo, An elementary introduction to the theory of the saddlepoint theory
of integration, Technical Report 55:2a, Research Laboratory of Electronics, M. I. T.,
May 3, 1950 (one section of Technical Report 55, On the evaluation of integrals of
the type f(T1 , T72**Tn)=
fF(s) exp[W(s, T1 ,
2
,
*,Tn) ds)
2.
M. V. Cerrillo, Transient phenomena in waveguides, Technical Report 33,
Research Laboratory of Electronics, M. I. T., Jan. 3, 1948.
3.
M. V. Cerrillo and W. H. Kautz, Properties and tables of Lommel functions of two
variables, Technical Report 138, Research Laboratory of Electronics, M. I. T.
(to be published).
4.
British Association Mathematical Tables, Pt. -Vol. B, The Airy Integral
(Cambridge University Press, London, 1946). Tables: (i) Ai(x) and Ai'(x) for
x = -20(. 01)2, and first fifty zeros and turning values; 8D, 6 2. (ii)Bi(x) and
Bi'(x) for x = -10(. 1)2. 5, and first twenty zeros and turning values; 8D, 62.
(iii) log Ai(x) and Ai'(x)/Ai(x) for x = 0(. 1)25(1)75; 7-8D, 62. (iv) log Bi(x) and
Bi'(x)/Bi(x) for x = 0(. 1)10; 7-8D, 62. (v) envelope and phase functions for both
Ai(x) and Bi(x) for x = -80(1)-30(. 1)2. 5; 8D, 62.
5.
G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University
Press, London, 2nd ed., 1948). Tables: (i) J 1/ 3 (x), Y 1 / 3 (x), H 1 / 3 (x), and
Arg H 1 / 3 (x) for x = 0 (. 02)16; 7D. (ii) J 1/ 3 (x) by simple computation with values
in (i). (iii) Zeros of J 1 / 3 (x), J_1/ 3(x), and J 1 / 3 (x) ± J 1 /3(x); 7D; see also
pp. 188-190; 199; 320-324.
6.
K. Pearson, Tables of the Incomplete Gamma-function (Cambridge University
Press, London, 1946).
7.
The Computation Laboratory, National Bureau of Standards, Tables of Bessel
Functions of Fractional Order, Vols. I and II (Columbia University Press, New
York, 1948, 1949).
8.
Tables of the Modified Hankel Functions of Order One-third and of their Derivatives,
Annals of the Computation Laboratory of Harvard University, Vol. II (Harvard
University Press, Cambridge, Mass., 1945). Tables: h(z), h'i(z), h'(z) for
Iz = Ix + jyl < 6, with x and y increments of 0. 1, and zeros of these quantities,
where h, 2 (z) = (Z)1/3 H)j (2) (Z), and Z3 = (2/3)z3/2.
9.
P. M. Woodward and A. M. Woodward, Four-figure tables of the Airy function in
the complex plane, Phil. Mag. ser. 7, 37, 236-261 (1946). Tables: Real and
imaginary parts of Ai(z), Ai'(z), Bi(z), and Bi'(z) for z = x + jy, x = -2. 4(. 2)2. 4, y
=-2. 4(. 2)0; 4D, 62.
11
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m'
cc
0)j
C"
cc
v
cc
IC)l
tLo
0)
cc
v
C"
C'
e~
"i
1-4
c')
cc
0)
'mC
o
C
co
0
0
o
c
0o
0
0
0
cc
c"
C'
0)
t-
v
E0)
0)
.
"i
cc
cc
IC)
C'
cc
C-
')
ll4
'C9
IC
Ccqo o
m'
0
cc
c
c
0
0-
)
0)
0
)
cc
0)
c'i
c'"
cc
cc
cc
0
0-
c'
I)
vc
cc
cc
.-4
C"
C'1
L-
C"
,-4
cc
0
cc
0)
C
0.
CD~
,4
C'I
0
(D
0
ca
l
~-
C.
C)
CD
C)1-4
cc
bi
cc
l0
C
r
C)
0~
cc
C-
cc
C)
cc
cc
14
O)
c-
cc
0)
C'a
cc
l
cc
N
cc
o
o
'u
o,
cc
c
36
I
c
TABLE 26
k
Coefficients of Bk in the Series (I.2)
0
0.246162705
1
.124427392
3
- .136757058 x 10 - 1
4
- .345631644 x 10 - 2
6
.151952287 x 10- 3
7
.274310829 x 10-4
9
- .703482810 x 10 - 6
10
- .101596603 x 10 - 6
12
.177647174 x 10-8
13
.217086759 x 10 - 9
15
- .281979642 x 10-11
16
- .301509388 x 10- 1 2
18
.307167366 x 10-14
19
.293868800 x 10-15
21
- .243783624 x 10-17
22
- .212026551 x 10-18
24
.147212333 x 10 - 2 0
25
.117792528 x 10 - 21
27
- .699013928 x 10-24
28
- .519367407 x 10- 2 5
30
.267821428 x 10-27
31
.186153193 x 10-28
37
11"-1-----
TABLE 27
Zeros of Ah 1 3 (B) for B real
k
Bk
1
3.3721344
2
5.8958433
3
7.9620250
4
9.7881268
5
11.4574233
6
13.0129143
7
14.4804371
8
15.8770394
9
17.2147133
10
18.5022978
11
19.7465442
12
20.9527565
13
22.1251975
14
23.2673582
15
24.3821434
16
25.4720025
17
26.5390256
18
27.5850140
19
28.6115343
20
29.6199596
38
rn
m
IBI
Fig. 3a
39
___
______
__
__
H
I
m
n
r
0.3(
4
IBI
Fig. 3b
40
7 A
52.5'
6'
m
n
r
llvv
IBI
Fig. 3c
41
______1_1___1____1___I_
____
3.
2.
2.
2.
82.5
I.
1,0
0.
0.
IBI
Fig. 3d
42
I
-m
z4
0
2
4
3
IBI
Fig. 3e
43
___I
I_
___
______
I
_
__
20
3
IBI
Fig. 3f
44
°
7z
n
zm
Q
4
IBI
Fig. 3g
45
111
__
- 0.41
m
cr
-1.6
0
2
IBI
Fig. 4a
46
3
4
.
a
n
c<
-1
-1.4
0
I
2
3
4
Fig.4b
Fig. 4b
47
--
.I
a
2.8
I
I /
2.4
2.0
97.5°/
I
-
a
90°
1.6
82.5
/
1.2
750
0.8
/A
0.4
I/////
//////
ldvl
i
-67.5°
I---600
_,
t
I
0
I
I
I
2
IBI
Fig. 4c
48
-
--
I
3
4
m
-
cr
<{
-o
2
3
4
IBI
Fig. 4d
49
___
I
I_____
1
3.2
--
2.8
--
120oX
2.4
1/ /
--
R
127.5"
\
Li
--
2.0
135 ///
I /
'n
(
tn:
/
142.5?
--
1.6
/
I
4
I50
I/
A:7V
Is
/,
/
s
Zl/
/
M
--
1.2 _-
)
X7
0.8 _
/
1/'
/
0.4 _
/
n
v
A
0
7
3
2
181
Fig. 4e
50
-------
I
a
A
m
CD
:
a
3
2
0
IBI
Fig. 4f
51
__
I
4
O
\
Sfr
m
~~~~~~~~·
~~ ~ ~ ~ ~
o~~~~
m,
5
or
~ ~ IPv
~~ ",~ '
no
,IL
'
-J
/ IT
~~~P
c~~JSo
~
4nZ ~
~
us
~
~-
I
'i~l~
~
o~l
0
~
~~
.
o
D
P~~~~~~~~~~~~~~~~~
c.-.
T
k
52
I-4
APPENDIX I
APPLICATION OF THE EXTENDED AIRY-HARDY INTEGRALS TO THE
EVALUATION,
COMPOSITION,
AND MECHANISM OF FORMATION OF
TRANSIENT PHENOMENA OF THIRD ORDER
The material presented in Appendices I and II is intended to show an important
application of the extended Airy-Hardy integrals to the evaluation and interpretation of
transient phenomena in dispersive media when the controlling saddlepoints are of third
order.
A general discussion of this subject is found in reference 1; in fact, these two
appendices are a reproduction of the material, with a few changes, of Chapter IV of
reference 1.
IV. 1
DEFINITIONS
IV. 10 DEFINITIONS OF PURE TRANSITIONAL TRANSIENTS.
Pure transitional tran-
sients are given, by definition, by
X
eW(s, t) ds
We shall start with the discussion of mixed transients, in which the effect of F(s) is
considered.
We shall first study simple typical cases which will help us in the attack
of the general problem.
IV. 11 THE COINCIDING POLE MIXED TRANSIENTS OF THE THIRD ORDER.
Let
F(s) be reduced to
F(s)=
1
1(IV. 1)
c
The denomination "coinciding" simply indicates that the pole is at s = s c , which is the
point of confluence.
The relation 1(I. 3) allows us to treat only the case for the contour yl which generates the Ahl(B) functions.
The corresponding integral for the coinciding pole transient of the third order is
given by
a+b(s-s) + c(s-sc) 2 + d(s-sc) 3
- s)
ds = e2
f(t)(s
'r
Bz z3
J
dz
Y1
when the transformation s - sc = z/
- c/3d is introduced.
We shall study in some detail the canonical integral, which reads
53
__
___
2(IV. 1)
Bz + z 3
41,lp (B) =
Zzi
L
e
3(IV. 1)
dz
Z
*
In order to integrate 3(IV. 1)one uses the auxiliary integral
B
Bz
jBeBz
dB = e
4(IV. 1)
z
Then
3(IV.
1)becomes
Then 3(IV. 1) becomes
fZ 3
3
=(B)
l
27i
z
B
3
1
eB
eZ dz
2ri
z+
3
eZ dz
dB = -1
30
Ahl
Z-
0
Y1
B
Y1
3
(B) dB
5(IV. 1)
since
Ahl,
3
I
(B) =
1
ri
e
Bz + z3
dz
1
The first integral o:f 5(IV. 1) has a pole at z = 0.
is needed
3
1
f
ez
z
3
e
dz =
Figure
ioo
+r/3
dT
l
ee
E1T +
-
(IV. 1) shows the contour which
T
d
1
d
e
-
i
T
d
1
6(IV. 1)
3
Then, expression 5(IV. 1) becomes, by using 6(IV. 1) and 3(I. 1),
l p(B)
=1oo
v+1
~
V+)! B
+
sin(3 (vl)
0
7(IV. 1)
1
l
='3 ~I
+
B_
B
r()
4
B
+
]}
..
which is the corresponding solution of the coinciding pole transient.
Curve 2 in Fig. 2a(IV. 1) shows the function 41 (B) for B real.
(In Figs. 2a, 2b,
and 2c(IV. 1) the point s c was selected as the point sd of confluence of the 2 saddlepoints
which correspond to the third-order case.)
The complete pole transient can then be written as
fp(t) = e A
p
p(B)
8(IV. 1)
~1,
54
_
_
Fig. l(IV. 1)
z PLANE
ILet F(s) be reduced to
IV. 12 THE COINCIDING ZERO TRANSIENT.
F(s) = s-
s
9(IV. 1)
C
The coinciding zero transient is written as
f
fo(t) _
1
1ri
f
a+b(s-sc) + c(s-sc ) 2 + d(s-sc ) 3
(s-sc) e
ds =
e
A
(fd
2
2
1
ze
Bz + z
dz
rri
1
10(IV. 1)
with canonical form
1,0 (B)
ze
2-i
Bz + z
3
11(IV. 1)
dz
Y1
This integral can be computed immediately.
It can be seen that
I1
i
ze
d
Bz + z3 dz
1
e Bz + z
dB
1
dB Z1ri
dz = dB
d Ah 1,
(B)
1
since the function converges uniformly with respect to B.
Hence in accordance with 20(III. 2) one gets
o
0
r
V+1\
k1(1)
(B)
C
B(V-l ) sin
(v+1)
0
12(IV. 1)
)
1
21T
r(4)
\32 B 2
2!
J3
55
_
IL__
-
r()
3!
B 3 +.
Curve 3 in Fig. 2a(IV. 1) shows the corresponding function 12(IV. 1) for B real.
The complete zero transient is given by
f0 (t)
A
e 2
=
1
IV. 13 COMPARISON OF PURE, COINCIDING POLE,
TRANSIENT (third order).
13(IV. 1)
(B)
AND COINCIDING ZERO
As a matter of illustration of the waveforms corresponding
to the pure, coinciding pole, and coinciding zero transient, we offer Fig. 2a(IV. 1) which
is computed for only B real. Curve 1 shows the pure transitional transient.
This is the case when
(Small shift case).
IV. 14 SHIFTED POLE TRANSIENTS.
s - sb
sb
Here we shall study the case of small shift, that is, sb is close to s c
sc
14(IV. 1)
.
The corresponding integral reads
f
(t) =
a+... +d(s-sc
a+
(
(S
1
)3
ds
3
15(IV.dz
5sd
eA
=
I)
15(IV. 1)
'1
where zb = s b - Sc; Zb is small.
The corresponding canonical integral now is given by
eeBz
= I+ j:
z- b
dz
16(IV. 1)
1
Now let us introduce the transformation
17(IV. 1)
Z - zb = u
We obtain
Bz + z
3
=u
3
2
2
3
+ 3u 2 Zb + 3uz b +b Zb + Bu + Bzb
or
zb+BZ
Pb 2ri
J,
u +u (B+3zb)
3u z
u
e
Let us introduce the new variable parameter
tIn Fig. 2(IV. 1) the variable v is used instead of z.
not change the results.
56
This change in notation does
X = B + 3b
(for small values of zb, X
.*
19(IV. 1)
B).
Before performing the integration 18(IV. 1) it is convenient,
for reasons of further
simplification, to substitute this value in 15(IV. 1) so that we shall first consider the
integral
l,p (x)=}
U3+uX
u
j
euue
3ugz
3
u
20(IV. 1)
Zbdu
1
for small shifts.
3u2 z
UZb
e
b
3
2
1 + 3u Zb
2 I(IV. 1)
so that 20(IV. 1) becomes
)( X
,p
(X)
-j
1
euX+u
du +
1i
d3
Zb 2ri
The first integral is a coinciding pole transient.
sient.
jue
uX+u
du
22(IV. 1)
The second is a coinciding zero tran-
Then, the first theorem of composition is given by
"The shifted pole transient is equivalent to the sum of a coinciding pole transient
and
3
b times a coinciding zero transient."
X = B +
there is practically no shift if zb is
Fig. 2b(IV. 1).
Since
3
small.
b
The function 22(IV. 1) is plotted in
The curve 0 is the coinciding pole transient.
coinciding zero transient.
Figure 2b(IV. 1) was computed for X real,
Curve 1 is the final shifted pole transient for zb = -0.
curve for zb = +0. 1.
The lower curve is the
zb = ±0. 1.
1, curve 2 is the corresponding
It can be noted immediately that
"In transitional transients,
shown here for the third order, a small shift of the
coinciding pole produces a strong effect on the wave envelope overshoot, and leaves
the rising part of the wave practically unchanged."
IV. 15 SHIFTED ZERO TRANSIENTS.
This is the case when
F(s) = s - sa
Sa #
The shifted zero transient is given by
57
_
_____
I
c
23(IV. 1)
I.
6(w
h
N
to)
G
I1
<t o*
l
I
$
t..
~
~~~CDo
00
a:
~
30
OLI
s
|
:
e
*
szI
i'
-
_
3
. -' -~ -I¢ -I· -I'~
,,,- ,, ~
58
i
fa(t)
a+(s-sc)b+. . . +d(s-s )
f
a+(s)b+.
.
a) e
(s -
+d~ ss)ds
a+(s-sc)b+. . . +d(S-sc )3
ai
Z-11 ((s
(s-s -
ds
SC) ee
)3
a+(s-sc)b+. .. +d(-sc
f i (Sc
Sa)
eA
(r
ds
e
ze
J
Bz + z
1
dz -ri
"~~'"
Z
e
dz
24(IV. 1)
'··i~
The first integral is a coinciding zero transient, the second is z
a
times a pure tran-
sitional transient.
Figure Zc(IV. 1) shows the plot of
Bz +
3
z
Za
a
jze
fi
2
Bz + z
e
3
jar
for za = 0. 01 and B real.
In Fig. Zc(IV. 1), curve 9 is the coinciding zero transient.
transitional transient.
Curve 0 is the pure
Curve 8 is the composite transient for a shift of za = -0. 01
and curve 10 shows the effect of za = 0.01 zero displacement.
It can be noted that
"In transitional transients (here proved for the third order) a small displacement
of a zero from the coinciding position has a very small effect in the transient wave."
IV. 16 DIPOLE TRANSIENTS.
This is the case where
s
F(s) =
- s
a
s - Sb
25(IV. 1)
since
F(s) =
s - sb + sb - sa
s - sb
1 +6
s - Sb
so that
a+... +d(s-s
fd(t)
)3
a+. .. +d(s-sc) 3
(s b -Sa)
ds + 2~i
e
s - Sb
2·r
The first integral leads to
59
_
_
I_
_
__
_
ds
26(IV. 1)
A
ds = e Ahl(B)
3r
e
27(IV. 1)
1
a
The second, in accordance with 20(IV. 1), leads to
(Sb - Sa)
2-ri
J
)3
3+d(s-sc
a+.. .
e
s - Sb
(
dsds =
e A+Bzb+z3)
= e(
b b
+ 3z
3b
2
1J
i
ue
r
uX+u3
J
uX+u 3
e e
du
U
du}
28(IV. 1)
which is the composition of a coinciding pole and a coinciding zero transient.
The final dipole transient then reads
IA
fa(t) = e
Bz +Z3
Ah (B) + e b
[6,
eA{
'
p(X) +
3
6Zbl, O(X)]}
29(IV. 1)
when
X= B+ 3z2b
The functions p(X)
1
B
30(IV. 1)
are given by 7(IV. 1) and 1Z(IV. 1), respectively.
Figure 2c(IV.1) shows the dipole transient composition that corresponds to the values
za =
0.2
Zb = ±0. 1
from which the following cases can be separated.
Curve
0J
G
Zb = +0. 1;
z
Z b = +0. 1;
z
Zb = -0. 1;
z = +0.2
a
Zb = -0. 1;
z = -0. 2
a
a
a
= +0.2
= -0.2
60
_
APPENDIX II
COMPARISON OF SECOND- AND THIRD-ORDER SOLUTIONS
IN THE CASE OF COINCIDING POLES
It is of interest to compare integral solutions corresponding to the contribution of
the second- and third-order saddlepoints in the case in which the saddlepoint orbit runs
over a pole.
Graphical comparison of the corresponding waveforms reveals at once the
character of both types of transient formation.
1.3
-
-
PURERESONANCE
FORMATION
TRANSITIONCOINCIDING
POLE FORMATION
- -._
.
1.2
1.1
0.9
---
2[F1E,rf(X.'i
POINTOF STEEPEST
TANGENT
0.6
GROUPVELOCITY
POINT
0.3,
0.2:
SIGNALPOINTS
O
-4
-3
-2
-I
0
.l
I
2
I
3
I
I
4
I
5
I
6
I
7
I
8
I
9
I
10
I
II
I
12
X -
Fig. l(IV.2)
Figure 1(IV. 2) illustrates the graphs of the second- and third-order coinciding pole
formation. Normalized units have been selected in order to give a proper basis of
comparison.
It can be observed that both formations are characterized by the rapid increase of
the response around the normalized point X = 0. For negative values of X, both cases
show a slow monotonic increase reaching the signal point at practically the same value
of X (X = -1). After this point, the second-order solution rises faster than the thirdorder solution. Both waves tend to the final state by oscillation. The second-order
solution shows faster oscillation and smaller overshoot value, and converges faster
than the wave corresponding to the third-order case.
The graphs of the third-order coinciding pole transients, illustrated in Fig. l(IV-2),
correspond to the functions associated with Ah 3 l1 The cases associated with Ah 3 2
and Ah3 3 are not shown.
61
__
A