Closed-form evaluation of the wave potential due to a spherical current source distribution

Closed-form evaluation of the wave potential due to a

spherical current source distribution

Citation for published version (APA):

Boersma, J., & Doelder, de, P. J. (1979). Closed-form evaluation of the wave potential due to a spherical current source distribution. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7911). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1979

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, I

. ;

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of ~~thematics

Memorandum 1979-11

October 1979

Closed-form evaluation of the wave poten~ial due to a spherical current source distribution

by

J. Boersma anQ P., J. de Doelder

Eindhoven University of Technology Department of Mathematics

P.o.

Box 513. Eindhoven The Net...'lerlands

Closed-form evaluation of the wave potential due to a spherical current source distribution

by

J. Boersma and P.J. de Doelder

1. Introduction

-r -r

Consider a current source distribution with density function J(r), having a finite supportin the volume V. Then the resulting electromagnetic field

-r

is expressible in terms of the wave potential ¢(r) given by

(1. 1) v -jkR e -r - - - dr' 471R R -r -r

stands for some rectangular component of J(r') and k is the wave number. It is 'Nell known that. (t,ri) satisfies the Helmholtz equation

=C~;J

,

...,.. r .: _Ii

(1. 2) M+ k2¢

-r

,

r i V

and the radiation condition at infi~ity (for the exp(j~t) time convention) . -r

The present note deals with the closed-form evaluatio~ of ~(r) in the case of J(i) having its support in the s!",herical vol'...lIlle

:;

1

~ a. Introducing

-r -r

spherical coordiantes r = (r,e,~) ana r' = ( r' ,8' ,(pi), we then rewrite (1.1) as (1. 3) <P(r,8,·:p) a

r

J

o

dr' 71

f

o

d8' 271 -jkR j' dm' J(r',6',f!J ' ) e _471R (r,)2 sin6-'

o

where R [r 2 + (r') 2 - 2rr' cos ~ ~ (1. 4)

cos y cos 6 cos 5' + sin e si~ 6_{' cos} (~ - f!J').

In section 2 it is shown that a closed-iorm evaluation of ~ is possible for a source distribution J of the form

, I I,', (L 5) J(r,a,rp) - 2 -cos sin (mrp ) m

where m,n,p are integers subject to 0 ~ m ~ n, p ~ n - 1; p stands for the n

associated Legendre function. In sections 3 and 4 two specific examples of source distributions (1.5) are examined. For these examples which were taken from Lee and Law [lJ, closed-form expressions for ~(r,a,rp) are derived valid fov r ~ a, i.e. in the interior of V. Results for ~(r,a,cp) when r ~ a might be derived in the same manner, however, we shall not go into the actual calculation. As a check the second example (section 4) is also treated by a different and independent approach based on the solution of the Helmholtz equation (1.2). Some concluding comments, additional to section 2, are pre-sented in section 5.

2. Wave potential due to the source distribution (1.5)

The key fcrmula in the analysis is the following -addition theorem:

(2. 1 ) -jkR e 41TR CD _ jk 41T

I

n=O (2n + 1) j (kr) h(2)(kr)· n < n > n

I

m==O (n - m)! e:

m (n

+

m)! pm (cos a) pm (cos a') cos m(rp- cp'). n n

Here r

=

min(r,r'), r

=

max(r,r'), EO

=

1 e:

=

2 for m

=

1,2,3, ... ;

further-< 12' > ' m

more j and h \ ' stand for L~e spherical Bessel and Hankel functions defined

n n

by

(2.2) _J , ( ) - [ l 1 T 'T ( ' . ' (2) ( )

-Il

'

TT H (2) (z) •

Z - -;;-- U L Z I , n Z - -2 n+L

n LZ n+. n z .

The key formula (2.1) is readily obtained from the additi~n ~heorem for Bessel functions (2.3) -jkR e "liTR 00 jk \' 41T

L

n=:O (2n + 1) j (kr ) h (2) (kr ) P (cos y) n < n > n

(cf. Watson [2, form.11.41(9), (10)], Abramowitz & Stegun [3, form.10.1.4S,46J, Stratton [4, form.7.10(87)]), combined with the addition theorem for Legendre polynomials

'. I

,,',

3

-(2.4) p (cos y) = p (cos e cos 6' + sin 8 sin 8' cos(<p - cp')) =

n n n

L

m=O (n m)! m e: ?

m (n + m)! n (cos 6) pm (cos 6') cos m(<p -n <p')

(cf. Stratton [4,form.7.S(46)J, Magnus, Oberhettinger & Soni [S,p.239J).

The source distribution (1.5) and the expar.sion (2.1) are now inserted into

the integral (1.3) for ~(r,6,cp). We employ the orthogonality relations

2rr (2.5)

J

dcp'~_cos m(-:p - <p')

o

where 0

,=

1 when m

=

m',

a ,

mm mm n (2.6)

_J

_{d8 ' sin 8 '} pm n (cos 0 1 cos (m'IP') drp'

=

sin 2'Tf cos e:

a

_{mm , sin (mcp)} m

o

when m ~ m'i and

8' ) , pm n' (cos 9' )de' (n + m)!

J

pm(x) p n' m (x) dx _2n 2

a

T 1 (n - n:)! nn'

,

-1 n

cf. Stratton [4, form.7.3(16),(17)J, Abramowitz & Stegun [3,form. 8.14.11,13J.

a...,{.

Then the 8'- ~ <p'-integration car. be carried out and we are led to the

fol-lowing result for ~ :

a h(2) (kr ) I ' ) p+2 (2.7) ~(r,6,9)

=

- jk

f

j (kr ) .r dr' pm (cos 6) cos m<p n < n > n sin 0

'2'

r _{(kr') (r' )p+2}

=

- jk [ h' I (kr) ( _jn dr' + n 0 a h (2) (kr') (r' )p+2 + _{jn (kr)}

f

dr'J pm (cos e) cos n n sin m<p

,

r valid for r ~ a.

Now it is well known that the spherical Bessel a.nd Hankel functions can be

ex-pressed in terms of elementary functions, viz.,

(2.8) h (2) (2'.) n jn+l z-1 e-]Z n \' -,-,~nc:....,----::..;1..~~ . + 0).' L 9.! (n - Il.) ! 9.=0 '2" )-£ .. ]Z

',', - 4 -(2.9) _{jn (z)} ~[h(l)(Z) + h (2) (z) J = n n ,-n-1 -1 jz n (n !(' , (-2jz) -90 ~ J z e

_I

_{Q,! (n} + _-) _£)

.

_! + £=0 ,n+1 -1 -jz n (n ,Q,) ! (2jz)-1 ~

_I

+ + J z e _{Q,! (n}

_-

_{£) !} 9,=0

cf. Watson [2,form 7.2(1), (2)J, Abramowitz & Stegun [3,form.10.1.16,17J, Mag-nus, Oberhettinger & Soni [5,p.72J. Consequently,the evaluation of <1' in (2.7) amounts to the evaluation of elementary integrals

(2.10)

f

e ±j kr ' (r ' ) p+ 1-Q, dr ' , ,Q, = 0 , 1 , ... , n •

The latter integrals can be determined through integration by parts if p + 1 - 2,

is a non-negative integer, that is, if p is an integer with p ~ n - 1, as assumed at the outset.

The evaluation of (2.7) for generai n is discussed in section 5. In the next sections we turn to some specific examples t,aken from Lee 3.nd La,,' [1 J .

3. First example from Lee and Law [lJ

As a first example Lee and Law [lJ consider a current density given ty -+

(3.1) J(r) _1-27\sln r ,28 _sincpcoscp+ 3

,E.,

_'\' 2 _Sln , 2 6 cos 8 sin ~ cos w,

1 _, I

which will be contained in the spherical volume r , ~ ), i 7\

wave length. Since

(3.2) pO (cos 8)

o

1, P2 (cos 2 8) 3 sin

L

8, p_ 2 ~cos Ell .)

(ef. Stratton [4,Appendix IV J ), we :nay rewrite (3.1) as

2w/k denotes the

15 sin~ 9 cos 8,

(3.3) 1 - -3 1 ~ P?2 (cos 8'1 sl'ni_,2'",) (r, 2 D2

(co'" P)

1\ _ '" T 1 () ,II ~ 3 -.; v sin (2cp)

'.',

5

-By means of (2.7) we find for ~1:

(3.4) '" ( 8 ) - J'k [h

O(2) (kr)

"'1 r, ,cp

=

where the integration variable r' has been replaced by p for convenience. From (2.8), (2.9) we have (3.5) hence r , -1 JZ e -jz r jz + j -1, -jz e

2'

z e- sin z _z

(3.6)

J

_jO(kP)P 2 dp 1

_f

sin(kp)pdp

= -

1 [- kr cos(kr) + sin (kr) ]

k k3 0 0 A h (2) (k ) 2

₌₁

A 1 ') -jkr 2n - j

f

o

p p dp

f

'k k e- J PPdP k3 (kr - J e -k3 r r (3.7)

through integration by parts. In (3.7) it was used that

A = 2n/k.

Inserting these results into (3.4) we find

(3.8) ~1 (r,8,cp) -jkr e [-kr cos(kr) + sin (kr)

J

-jkr e 1 + 2j'IT (1 + jkr) sin(kr) + sin(kr) k3r 1 1 + 2jrr sin(kr) - - + k2 k2 kr 1 1 + 2jn - - + jo (kr). k2 ,( ,2

Consider next ~2(r,8,cp) which is to be determined from

(3.9) r

A

f

j2(kP)p3dP + j2(kr)

f

h~2)

(kp) p3 dPJ

o

r 2 • P (cos 8) sin (2cp) . 2 From (2.8), (2.9) we have (3.10) - JZ , -1 e-jz [; _ • ~,-1 ~JZ - 3 -2z

_J

,

*)

(3.11) hence (3.12 ) (3.13 ) j~(z) =Re L 6 -sin z 3 z 2 3 (3 - z ) - cos z 2 z 1 [8 - (8 - k2r2) cos(kr) - 5kr sin(kr)] , A

f

-

:12 (2) (kp)p 3 dp r r

f

cos(kp)pdp

o

1 4 2, 1 -i~r ,~ 2 2 d (8 + lOjif - 1 \ ) +"4 _e - '.:j + 5jkr - k r ). k - k

We then find after some elementary calculat~o~s

(3.14 ) .- 'k

. '-j -

J r , ) 2 2 4 4 ¢2 (:c, e, <p) = J 3" e i 3 {24j - 24kr - 8jK-r- -+-cos (k;::) [-24j - 4jk r -jk r ] A

Lk

'

r 1 1 2 2 4 4 8e-jkr 2 2

~)

[~(24 + 4k r + k r ) - _ 3 (3 + 3jkr - k r ) Tfk2 k r kjr

Nex't, consider ¢3(r,e,cp) which is to be determined from

This term can also be expressed as 8lh(2)

(kr)~

- 2

(3.15)

..,

_, 1'3 (r , 9, cp) = -

2 -

[h?) (kr) 10>" 2 .) ( j3 (kp)p4 dP + j3 (kr) / '

h~2)

(kP)p4 dP ] •

o

r 2 • ? 3 (cos 9) sin (2cp) •

Again from (2.8), (2.9) we have

(3.16) (3.17 ) hence (3.18 ) (3.19 ) h~2)(Z) -1 -jz [ 1 -1 -2 -3 z e

-

6jz

-

15z + 1 5j z ] ,

j3(z) =Re h (2) ₃ (. 2, \ sin ₄ z (15

-

6z2) + - -cos ₃ z (-15 + z ) 2

,

z z r 4 • r 22 1 : : : 23

J

j3(kp)p dp =~

f

sin(kp) (15 -Bk p )dp -~

f

cos(kp) (15p -k p ldp

o

~< 0 k- 0 1 -jkP'48 33 1 ~k2 2 - e ' + , J' Kp - ::I,' P - 5 \ k 3 J

It

- jk P ) r

Inserting these results into (3.15), 'lie find in a straightforwa:::d manner

(3.20)

, (7 20 .. 2' 2 2 c. k4 4

-J + I ~ r + c. r + k6 r ) e 6, jkr

J

\

8 -3

[

1 2 2 + 6k4r4 k6r6)

- - -

- - -

(720 + 72k r + 4'/ k2 k4r4 48e-jkr (15 + 2 2 ,,3 3)

*)

+ 15jkr - 6k r - JK r k4r4 + j(48 + 66jn - 36n2

The final closed-form expression for ~(r,e,~) is now obtained by addition of the results in (3.8), (3.14),(3.20). The expression for ~ is valid for r ~

A,

i.e., in the source region.

Lee and Law [lJ are especially interested in the numerical values of the deri-vatives

(3.21)

at the origin r

=

0; here x

1,x2,x3 are Cartesian coordinates. To evaluate these derivatives we expand ~1'~2'~3 in power-series in powers of kr, viz.,

(3.22) (3.23 ) (3.24) '" e --..L[2_J''''' _ 1 + 2jn k2r2 + (k44.1 "'l(r,,~) _{k 2} " 6 o . r ) J 3

[

-, 2

e

s~n sin (jl cos ~ , 48j 3 3 4 4 2 3 k3r 3 105 k r +O(k r ) +j(48+66jn-36n -8jn )~

sin2 e cos

e

sin cp cos (jl •

The second derivatives of

ter~s

of order k3r 3 and higher vanish at the origin; for example

9 -a2 4 a2 2 2 2 2 2 2 2 2

_o

ar r 0, r _{(xl + x 2 + x 3 )}

=

4(x 1 + x2 + x3) + 8x1

=

2 2 aX 1 aX1 3 , 2 (r s~n 8 sin ~ cos ~) aX13~ r 2 , 4 2 , 2 J rL1 + sin 8 -s~n 8 cos ~ s~n ~

o

at r

=

0, 32 (3 , 2 , ) 2 r s~n 8 cos

e

s~n ~ cos ~ 3x 1 0, etc.

Thus i t is sufficient to retain only the terms up to and including O(k2r2), yielding (3.25) ¢(r,e,(,o)

-t

_[2jrr k 1+2j'Tl k22 10+4jrr. 22 , 28 ' Otk33)J 6 r + 15 k r s~n s~n ~ cos <P + II r 1 [2' 1 + 2jrr 2 2 2 2 10 + 4jrr 2 3 3 k 2 J1T - 6 _{k (xl +x2 +x 3 ) +} 15 k x₁x₂ + O(k r )J All second derivatives I

mn 2

a ¢/ax

ax

at the origin can easily be determined m n now: (3.26) 1 + 2jrr 3 10 + 4jrr 15 ' 1₁₃ O. As a check we have ~ (3.27) 6~(r

=

0)

=

III + 122 + 133

=

-1 - 2jrr

in accordance with the Helmholtz equation(1.2). The magnitude and phase of 1

11,112 are found to be

2 ~

- 10

-(3.28) 1

11

=

2.1207 55044 exp(-j 99.043°), 112

=

1.0706 4606 exp(j 51.488°),

which should be compa~ed to the values of Lee and Law [lJ , viz.,

(3.29) 2.1327 exp (-j 100.13°), 112 1.0740 exp(j 50.73 ).

°

4. Second example from Lee aud Law [IJ

As their second example Lee and Law [lJ consider a current density given by

(4.1)

again contained in the spherical volume

l"tl

::;

A. Clearly,

J("t)

in (4.1) is of the form (1.5) with m

=

0, n

=

O. Thus we have from (2.7) that the correspond-ing wave potential ~ is given by

(4.2) 1>(r,E),cr) JO(kp){p . 2 -3 2 .2- + A \ 3 4 +jO(kr)

J

h~2)(k

P)

{p2-2f-+3~}dPJ,

r \ valid for r :;; \. By means cf (3.5) we determine (4.3) (4.4) r 2 3 k

b

sir.. (kp) {p - 2

t-

+ 3

~

2 }dp 2 - kr 'IT / . (2)(k.){ 2 ? 03 :)4, :\ : n . c: 0 - _ - -+ 3 - , d.)

= i

J

' 0 ~ . ).. '» k r . /, <. J~ -jkp, 2c2

~

e t p - - . - + } dp A \2 -J. ko _20-') _{' 3 '} ~ _{. 4J'p o·} 2 _{4 18 lS'}

1\

~ [ •. p + _ _ _ ~_ + 1. _ _ + -,]p P J ] , - A ) 2 k kA

k~

2 - k2 +

'K2~

2 - ,)(3,2 r K . " A · , 1\

through integration by parts. - 11 --jkr e

-

-k3 9j _ (1 + 2j ') 2 IT .. IT 3 1 3 3 ] --2 K r 4-:r

The present results are inserted into (4.2), thus leading to

9 )kr + 2IT2 (4.5) <P(r,e,ql) _ j k[=-j _e-:--_] k_r_ k4r ? 2 Q 1 2.2 J k3 3 - jkr-{-=- - [ - + (1 - -::-) kr - -;, k r- +"""2 r Je } + + 1 3 k r IT IT 2rr2 40-sin(kr) kr 2 3 -:rk r -jkr e

The latter result is now re-de=ived by a different and independen~ approa~h.

We observe that for the present example the wave potential <P is spherically

symmetric, i.e., <P

=

<P(r). Then we fi~d from (1.2) that <P(r) must satisfy

the Hel~1oltz equation

3

(

f)

2 , 0 $ r <

,

1\ I

(4.6)

, r >

A.

Moreover <P(r) must satisfy the radi~t~on condition at infinity a~d OCr) must

be finite at r = O. Equation (4.6) is easily solved, viz.,

(4.7) 9(r) sin(k~) A + <P (r) 0 5 r $

A,

kr 0 ' -jkr B e kr r 2: A,

where A and B are arbitrary constants, and <i>O(r) is a particular solution

- 12

-(4.8)

determined by trial and error.

The integration constants A and B are found by requiring continuity of ¢ and

¢' at r = A, i.e.,

(4.9) ¢(A - 0) ¢(A + 0), ¢'(A - 0) ¢' (A + 0).

Thus we obtain

(4.10) A j 1T (4 - .=.!I.. -6; -7 '- + - ) 9j B

k 2 1T n 2 2n3 '

Insert these values into (4.7), then the result (4.5) is recovered, valid for r $

A.

In addition we find

(4.11)

-jkr e

kr when r ~ A.

Lee and ~aw [lJ are interested in the numerical value of the derivative III

a2~

/

ax~

at the observation point xl

=

0, x

2

=

0, x3

=

0.4'/... Such a

deri-vative is obtained from

(4.12)

which reduces to

(4.13) r) 1 ¢' (r)

r

for an observation point on the x

3-axis. Through differentiation of ¢(r) in

(4.5), and setting r

=

0.4A, we find

(4.14)

, _6J'_9 9J' ?' ]

- ~ (4 - -2 + -2J kr) sin(kr)

k 3r 3 1T 1T 21T 3 - 1T r- 2 A

5. 125 = - - ( 1 32'TT4 13 -2 8,,) 25j §.i 9

2i

(4'TT) 125 + -16'TT (4 - 'if -

2

+ 3 ) cos 5 'TT 'IT _ 12Sj (4 _ 38j

_1-

+2L) sin (4'TT5) 2 5 'IT 2 2 3 64 'IT 'TT 'IT - 0.9273 7407 - 1.6017 7327 j or in magnitude-phase-representation (4.15) _{III (xl}

₌

_0,x2 1.8508 6474 exp( -j 120.069 ). o

This result agrees reasonably well to that of Lee and Law [lJ in their table III. A simpler result is obtained at the cbservation point xl

=

0, x

2

=

0, x3

=

~A, viz. , (4.16) I 11(x1 0, x 2 ~ 0, x3 ~A) 13 17

_i

(4 ~) -0.5713 2684 0.9829 7573 j. +

-

-2 'IT 2 2'TT 4 'IT 'if 2

Addendum to section 2

C::msider a cur::cent densit.y

-+ _{(E.)p pm} cos

(5.1) J(r) (cos 8l

sin (mcp) , a n

similar to (1.5), and contained in the spherical volume

1;1

~ ai again, m,n,p are integers subject to 0 ~ ill ~ n, p ~ n - 1. Let the corresponding wave

po--+

tential be denoted by ¢ (r). From a study of the specific results in (3.8),

pmn

(3.14), (3.20) and also (4.5), it is believed that in general ~ has the

pmn

following structure: If P - n is even, then

(5.2) 1 cos

sin (mcp)

where Qe(k2r2) is a polynomial in k2r2 of degree

~(p

- n), and A is a constant which depends on p,n and kai

if P - n is odd, then

-+

(5.3) <I> (r)

pmn

14

-where QO(k2r2) is a polynomial in k2r2 of degree

~(p

+ n + 1), and A,B are

constants which depend on p, nand ka.

-+

According to (2.7), <I> (r) can be expressed as

pmn j ~ h (2) (kr) kr (5.4) <I> (r) -+ =

f

_jn pmn _{k2(ka)P n} 0 ka h (2) (x) xp+2 dx] + j (kr)

J

p m (cos 8) n kr n n p+2 (x)x dx cos (mq» sin

where we introduced the new integration variable x = kr'.

+

An interesting special case occurs when p = n~- Then by use of the recurrence

relations [3, form.l0.1.23] (5.5) we have (5.6) d rr n+2 dx LX jn+1 (x)] = x n+2 jn (x), dxd -_ [n+2 h(2) (x)] x n+1 = xn+2h(2) (x), _n h (2) (kr) n kr

f

j (x) xn+2dx + j (kr)

o

n n ka

f

h(2) (x) xn+2 dx kr n (kr) n+2[h (2) (kr) j 1 (kr) n n+ j (kr)h(2)1(kr)] + (ka)n+2h (2)1(ka)j (kr)-n n+ n+ n _j(kr)n + (ka)n+2h (2)1(ka) j (kr) n+ _ n

by means of the well-known Wronskian relation [3,form.l0.1.31]. Thus we find

(5.7) <I> (r) -+

runn

For n = m = 0 and a =

A

the present result reduces to that in (3.8). Notice

that (5.7) has the structure conjectured in (5.2).

Next we reconsider the evaluation of the second derivatives of-<I> at the origin

-+

15

-by expanding 1> in ~ pcwer-series in powers of kr while retaining only the terms up to and including O(k2r2). From the estimates [3,form.10.1.4,5]

(5.8) j (x) =O(x n ), h(2)(x) =

o

(x-n-1) as x +0,

n :1

:.t is readily found that

h (2) (kr) kr p+2 ka h (2) (x) xp + 2 (5.9) { _jn (x)x dx _{+ jn} (kr)

f

dx n J _n 0 kr = O\(kr)-n-1)

o

((k:!:")n+p+3) +O«kr)n) -+ since p + 2 ~ n + 1. v~nish when n ~ J.

Conseque:1tly, all second deriv~tives of <i> at r

pmn

This leaves t.."le followir,g" t.l,ree cases to be considered ;

(i) n = 0, m = O. ThreE subcases are to be distinguished, namely,

a) p -1. Then 'tiE; have from (5.4),

kr ka

o

(5.10) -+

= _

j ka -h (2) I k ) 1> -1 ,0 I 0 (r) 2 L· 0 \.r k

f

jO(x)xdx + jO(k:!:")

o

kr

f

(2 ) hO (x) xdx] _ j,~a 1 2 K [ -' j - e -jk!:" kr kr

f

o

sin x dx + sin(kr) j kr ; j -jkr [ _ _ . J _ + e sir. (kr) kr -jka e ] kl.' kr -+ ka

f

e -jx dx j kr

The second derivatives of <i> do not exist at r 0 i:1 ~his case; this is

not surprising since J(;) = ar-1 is singular at the origi.':1.

b) P

=

O. 'I'his subcasoe h~s been treated in (3.22) and (5.7),vi2.,

(5. III 1 [1 + j (ka i 2 fl 1_( 2) (ka) {1 k2 1 ~ ? -"-,,L. _ [;, r " A " 'k'" -r. '. ] + L \. r ' ;

, ' - 16 -c) ~. Then we have frcr.l (5.4) , (5.9) , (5.12 ) (5. 13) , ~2 ka (2) 2 J [O«kr)P')+ jO(kr)

f

hI) .

(~)Xp+

dx] . 2(. )p 0 .< Ka The i:1tegro.l I P ka

f

e-jx xp+1 dx

o

may be determi~ed r.ecursively t t rough

(5.14 ) I

=

J '(k ' a ) ,p+1 - J.p '( + 1) I , P

P ~1 0,1,2, ... ; 1_1 j (e

-j ka - 1).

(ii) n = 1, m = 0 or 1. Two subcases are to be distinguished, namely,

a) P

=

,0. Then we have £rom (5.4),

(5.15) ell ( \ + . Qml r; m cos • P1 (cos 8) (m<p) sin +

"

Again t.'1e second jerivatives ::>f 01> do not exis"C at r v, e.g. -for m =

2 2 12 ,) ₃'") _{~ 2} x3 (;:2 + x3) :2 2~. (5.16) '\ 2 [r"'P1 \COS 6)] ~[x3(x2 +}:2+x3'

J

2 .: _A~)3/2 oX 1 oX1 (x 1 + h2 + -' / , 2 2

- cos 6(sin-e s~n Q + cos 6)

,

.

- 17 -

-+-depends on the direction in which r -+- O. Again this is not surprising,

since

J(i)

=

pm(cos 8) cos(mw) is not continuous at

i

=

0 (let alone

-1' S1."

Holder continuous).

b) P ~ 1. Then we have from (5.4), (5.9),

(5.17) ¢ 1 (r) -+ pm -; ,! ka

f

h '1 (2) ( x ) p+2 X \.. ~, •• .. ,

o

-+

All second derivatives of ¢ vanish at r O.

ka ]

f

-

,(

fi 2) ( ) p+2d l x x x

o

kr 3 31m cos ~ + O(k r ) J,P1 (cos 8) , (mrp). J s~n

( i i i ) n = 2, m 0, 1 or 2, p ~ 1. Then we have from (5.4), (5.9),

-+ (5.18) ¢ 2(r)

pm

j

t

_o

«'Kr\jP+2), ,ka (2) ( , _{P+2d ]} m 6) co, s (m )

+J2(krloJ h2 XIX x P2'cOS s~n q>

k20o.)P

ka 2 2

1

( (2)( \ p+2_d k r

r:

k3 3) m! 8'cos()

J h" XI X X --,-~- + ../ (. r

J

P ~ ,cos I ' mq>.

o

""

l:: L. s~n

Here one has from (3.10),

(5.19) ka ka

f

h '2 (2) ~XIX-. \ p+ 2d x -- J'

f

o

0 -J'x p-l P 0+1 e [3x + .3 j x - x · ] ax , [ r l ' I I ] J ~p-2 + -J p-1 P to be determined from (5. 14) . -+

The second derivatives of the pertaining l 's at r

tained from (5. 11), (5. 12), (5.:8).

ob-,

.

18

-References

[lJ S.W.Lee and C.L. Law,Singularity in Green's function anj it;;; numerical

evalua~ion, Electromagr .. Lab., U:1iv. of Illinois, Tech.Rep. Etv!79-10,

June 19/9; a revised versi.on of the report, co-authored by J. Boersma

and G.A. Deschamps, will appear in IEEE Trans.Antenn~s Prop.

[2J G.N. Watson, A treatise on the theory of Bessel f'.lnctions, Cambridge

Univ.Press, Cambridge, 1958.

C3J M. Abramowitz & I.A. Stegun, Handbook of mathematical functions, Dov~r.

Publ., New Ycrk, 1965.

[4J J.A. Strattcn, Electromagnetic theory, McGraw-Hill, New York, 1941.

[5

J

W. Magnus, F. Oberhettinger e, R. P. Son1, FOrIDLllas and theorems for t:.he