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LJ

NEW YORK UNIVERSITY

â– pOURANT INSTITUTE - LIBRARY

1251 MÂ«rcer St. New York, N.Y. 10012

/0^Â». ^x T^. NEW YORK UNIVERSITY

^ I ^ lY ^ Institute of Mathematical Sciences

0-1

Wl

s Division of Electromagnetic Research

cccxx^

'^^crr yX+

RESEARCH REPORT No. BR-7

Determination of Coefficients of Capacitance of Regions

Bounded by Collinear Slits and of Related Regions

B. EPSTEIN

CONTRACT No. AF- 1 8 {600)-367

AUGUST 1954

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

RESEARCH REPORT No. BR-7

DETERMINATION OF COEFFICIENTS OF CAPACITANCE OF REGIONS

BOUNDED BY COLLINEAR SLITS AND OF RELATED REGIONS

by

Bernard Epstein

/j. Q^-d-^^IL^

Bernard Epstein ^

P^f^^EW YORK UNIVERSITY

. POURANT INSTITUTE - LIBRARY

Morris Kline 25( Mercer St. New York, N.Y 10012

Project Director

^i^

The research reported in this document has been made possible through

support and sponsorship extended by the Office of Scientific Research,

H. Q. Air Research and Development Command, U. S. A. F. , Baltimore,

Maryland, under Contract No. AF 18(600)-367. It is published for tech-

nical information only, and does not necessarily represent recommenda-

tions or conclusions of the sponsoring agency.

New York, 1954

C2L

Abstract

A number of formulas are derived for the coefficients of capacitance of

a domain which consists of the entire plane minus any finite number of col-

linear slits. It is shown that any of a broad class of domains possessing

a certain symmetry property is conformally equivalent to such a 'slit -domain',

and that the problem of determining the conformal correspondence is essential-

ly one of mapping simply -connected domains rather than multiply-connected ones.

As an illustration, the coefficients of capacitance of a 'bi-filar shielded

cable' are computed.

Table of Contents

Page

1, Introduction 1

2, Regions Bounded by Collinear Slits 2

3Â» Extension to More General Regions 6

U, An Illustrative Example 8

References 12

- 1 -

1, Introduction

In the study of electrostatic field problems the principal objective

usually is to determine the potentisil and its gradient (the field strength)

throughout a given domain boimded by a system of conductors. Frequently,

however, it is necessary only to determine certain constants of capacitance.

In this paper vre consider the latter problem for a certain class of plane

domains.

Let n be a domain consisting of the entire x-y plane with any finite

number of slits along a single line. Several formulas for the coefficients

of capacitance of such a domain are derived, two of which appear to be well

suited for nu'nerical computations. One of these formulas is based on the ex-

plicit representation of the potential as the i^eal part of an analytic function I- J*' - '

while the other formula has the feature of requiring a knowledge of the po-

tential only on the line containing the boundary components of Dj it does

not involve any derivatives of the potential, A convenient method for de-

termining the field along the line containing the boundary components has

[2]

been givÂ«n by the author in a previous paper, -^

Since the coefficients of capacitance of a domain are invariant under

conformal mapping, the formulas which are derived may be employed to compute

the coefficients of any domain which can be conformally mapped upon a domain

D of the type described above. This procedure can be applied to a certain

class of domains which are of practical interest. In these cases the mapping

problem involves essentially only simply-connected domains rather than multi-

ply-connected domains. One particular case of interest, the 'bi-filar shield-

ed cable' is considered in some detail, and as an illustration of the pro-

cedure, the coefficients of capacitance are evaluated numerically for one

such domain.

- 2 -

2, Regions Boxinded by Collinear Slits

We consider here the problem of determining effectively the so-called

coefficients of capacitance of a domain D consisting of the extended (x,y)-

plane with a finite number, m, of collinear slits, cut along what may be assumed

to be the x-axisÂ»

We recall the definition of these coefficients:

/ Su.

(2.1) Pik^-cT aJT^s, (i, k = l,2, - ,m),

'^k

where u., the harmonic measure of the i-th boundarj'^ component, is the harmonic

function whose boundary values are unity on the i-th component and zero on all

other components (the existence and uniqueness of this harmonic function is

well known from the theory of the Dirichlet problem); C, is any curve, described

in the positive sense, surrounding only the k-th component; and â€” indicates

differentiation in the direction of the outward normal. As follows immediately

from the C auchy-Riemann equations, the p., may be defined alternatively as the

increment in the harm.onic f\incticn v. conjugate to u. which results when C, is

described cnce in the negative sense.

We also recall several important properties of these coefficients of cap-

acitance:

i = l,2,...,m;

(2.2a)

Pik ^ Pki 5

(2.2b)

r Pik - 0^

k=l

(2.2c)

Pii > ;

(2.2d)

Pik < ^^

i / k.

The p., are, of course, conformal invariants. Hence we may assume that D con-

sists of the entire plane minus a finite niomber of slits lying on the x-axis,

one of which extends to infinity in both directions (see Fig. 1); this con-

figuration can always be realized by a suitable inversion. It will be seen

- 3 -

that this assxunption eliminates the possibility of any convergence difficulties.

For brevity such domains will be called slit-domains. We number the finite slits

1, 2, â€¢â€¢â€¢, m-1 from left to right; and the infir.ite slit is the m-th.

y

Figure 1

-CD

Or

^2 ^3

m-i

a b

m m

+00

We proceed to derive various formulas for the quantities p., which may

prove useful for numerical computations. First we sha32 employ the definition

(2.1); later we shall derive a formula based on the alternative definition

given following (2.1),

Taking into account (2,2a) and (2.2b) we see that it suffices to determine

the p., with i < m, k < m. Let f . (x) = u.(x,0), so that, in particular,

f . (x) = for xb. By the Poisson formula we have, for any point

off the X-axis:

(2.3)

m f. (f)d4

I 1 / m f. (f

u (x,y) Â» M / ^â€”

T-T

a^ (4-x) +y

We take as the curve C, a rectangle with vertical sides passing through the

gaps (a, ,b, ) and (a, -i *ti, ,), It is easily found from (2.3), by differentiating

under the integral sign, that for any value of x:

^^i I 2A

dTl- ~ ' ^ " maxda^Mb^l)

(2.U)

l^^il 2A^

aT"! - ~3 '

' ' ny

Tiy

(one uses, of course, the fact that |f . (4) | 5 !)â€¢ Hence as the horizontal sides

recede to infinity the contribution to the integral (2.1) from these sides

vanishes, and therefore (2.1) may be written as follows:

(2.5)

ik i,k ~ i,k+l *

where

(2.6)

- U -

\i 'jC ^ dy, a. < X < b.

(of course p. -I. -I.,).

^ ^im i,m 1,1

Ws wish to rewrite I. .in such a form as to involve only the values of

i,j

au.

u. on the x-axis, i.e., the values f.(x). If one expresses ^ â€” by differ-

1 1 ox

entiating (2.3), inserts this expression into (2.6), and interchanges the order

of integration, one obtains the following expression for I. .:

(2.7)

P /m f (Â£)d4

I. , - - P / -ip , a. < X < b..

Here the symbol P denotes the Cauchy principal value of the integral.

A second expression for I. .is obtained by the follovring artifice:

Since the right side of (2.6) gives the same value for any choice of x in the

j-th gap, we integrate both sides of (2.6) over this gap. Then we obtain

'V^j>^i..i

b . .oo 3â€ž

'a . '^-00

aT- ^y^-

Interchanging the order of integration [this is easily justified with the aid

of the first inequality in (2.U)J we obtain

.oo

Â«oo h

(2.8) (b -a )I

J J i> J

^-00 a .

BU^

aF

CO

n 1^-00

|y|

m

Ih

< u(b ,y)-u(a.,y)[. dy

^(4) r ^-^ ^-j 1

^ K4-bJ V (4-aJ V J

d4 ydy.

Now we interchange the order of integration once more, and obtain the

following expression for I. .:

m

(2.9)

^i,j i^(^r^y^

f^U) In

4-a,

4-b

d4.

A third expression for I. .is obtained as follows. Since f.(4) is con-

1, J 1

tinuous and f.'(4) and In

- 5 -

^-^J

are absolutely integrable, an integration by

parts enables us to rewrite (2.9) in the form:

b

(2-lÂ°> hS iT^ / "q(f.)[(4-aj)lnK-a.|-(*,-bj)l.lS.b^|]dÂ£i

the integrated term vanishes since ^^^(^1) = -^i^^m^ " ^'

Equations (2.7), (2.9), (2.10), together with (2.5), give three formulas

for the coefficients p., which employ only the values of u. on the x-axis.

These equations were all derived using the definition (2,1) of p , . A fourth

formula is obtained by using the alternative definition, as follows. It is

shown in [l] , Â§ 91 that the analytic function v^{z) = u^+i v^ must satisfy

the condition

P,(z)

(2.11) wj(z) ^

i^ ' /^n

/. TF (z-a )(z-b )

where P.(z) is a polynomial, of degree not exceeding m-2, with ra-1 real co-

efficients which are uniquely determined by the conditions :

/j r +1, for j - i,

(2.12) / w.'(z)dz = { -1, for j Â« i+1,

â– 4. ^ 0, for ;j ?< i, j / i+1

(It is shown in reference [l] that of the m conditions (2.12) only m-1 are

independent.) Now employing the alternative definition of p , and taking

for C, the doubly-counted k'th interval, one easily obtains from (2.11) the

k

result

A+1 P.(C)dC

(2.13) Pik-^Vn

m *

/TT (4-a,)(?-b )

where the ambiguity in sign is most easily resolved with the aid of (2,2c)

and (2. 2d).

â€¢* It must be remembered that the denominator of w^(z) changes sign on

successive intervals (a., b.).

- 6 -

Insofar as numerical computations are concerned, it would appear that

formulas (2.9) and (2.13) are especially suitable. While (2.13) has an ad-

vantage over (2.9) in that integration is necessary only over a single interval,

it involves the solution of the system (2.12) for the coefficients of the poly-

nomial P.(z) and this may become laborious for large values of m. In this case

it might prove preferable to employ (2,9), obtaining the fvmction Â£^{x) to the

desired degree of approximation by the method given in [2^, Â§ 6.

3, Extension to More General Regions

Since the coefficients of capacitance are conformal invariants, they may

be determined by the method described above for any domain which can be mapped

confomially onto a slit-domain, A simple example of such a domain is the entire

(x,y)-plane slit along a finite number of arcs of a circle (cf, [l] , Â§ 92), By

a suitable linear transformation we can map the circumference of this circle

onto the x-axis, thus obtaining a slit-domain.

Another domain to which this method applies is a domain bounded externally

by one circle and internally by two others. By a suitable linear transformation

such a region may always be mapped into a domain with the center of all three

circles collinear (cf. Fig. 2), y

Figure 2

- 7 -

Now suppose that the upper half of this domain, which is simply-connected,

is mapped conforraally by an analytic function Y, ' f(z) upon the upper half of

the ^-plane. Then the upper halves of the three circles are mapped into seg-

ments of the real axis of the )^-plane, and the three segments of the axis of

s yi i u t i etry (AB, CD, EF) are mapped into the remainder of the real axis. By the

Schwarz reflection principle, the entire domain of FigÂ» 2 is then conformally

mapped onto the ^-plane with three slits along the real axis, i.e., upon a slit-

domain, A particular case of practical interest is the 'bi-filar shielded con-

ductor'. This case will be discussed in some detail in Section U,

Closely related to the configuration of Fig. 2 is that of the plane with

any finite number of circular apertui^s whose centers all lie on one line

(cf.Fig.3). Each half of this region is simply-connected. If, as before, we can make

O

^ i-^v^

1-2 ya.8

of the real axis. Thus the domain of Fig. h has been approximately mapped

into a slit-domain, and the formulas given there may be employed to determine

the p^j^.

- 11 -

To give a numerical illustration, let a - pQ , P â– r. Then p Â«â– ^, and,

since the infinity products (U.U) and (i;.5) converge rapidly, one obtains very

easily

l.f(iliJ^ ,p)

(U.6) ^^ - 0.212, ^ = U.726, "-^ >g - 8.628 .

1-2 y^

In order to work with more convenient numbers, we employ the transformation

(U.7) z - -^ z,

thus obtaining in the z-plane a slit-domain characterized by the following

numbers (cf. Fig. 1):

(U.8) a^-= -8.1, b^= -U.5, 3^= -0.2, b2= 0.2, a^= U.S, b^- 0.1.

On account of the low connectivity of the domain (m=3) it was decided to era-

ploy formula (2.13) to evaluate the p., . The coefficients ppp and Pp-, (the

numbering of the boundary components is given in Fig. U and Fig. 5, and is

in accordance with the numbering given preceding Fig. 1) were determined as

follows; The polynomial Pp(z), which according to the discussion of Â§2 is

2

of first degree, was replaced in (2,12) by the quadratic polynomial az +bz+c

with unknown coefficients a, b, c. The integrals appearing in (2,12) were

computed by interpolation (the Tchebyshev five-point formula was used) and

the resulting system of linear equations was solved for the coefficients a,

b, c, with the following results:

(1+.9) a = -9.3 X 10"^, b - -1.962, c - -11.608.

(The radical appearing in (2,12) was taken positive on the first and third

intervals and negative on the second interval.) The extremely small value

obtained for a, whose exact value must be zero, ser*ves as an excellent check

on the accuracy of the computations. The expression for Pp(z) thus obtained

- 12 -

was then employed in (2.13), with the following results:

(li.lO) P2^ - -2.153, P22 - +3.820,

3y symmetry, p^, â– Poo> ^^"^ ^'^^ ^^^ remaining six coefficients are obtained

with the aid of (2.2a) and (2.2b). Thus, the following table of values of

the p., was obtained.

2

3

+3.320

-2.153

-1,667

-2 ,153

+3.320

-1.667

-1.667

-1,667

+3.33U

As a check, p-p was also computed by means of (2.5) and (2.10), and excellent

agreement was obtained.

Acknowledgment

We wish to thank Mr, Charles Kahane, who so capably carried out the

computations connected with the illustrative example of Section U,

References

[1] N, I, Muskhelishvili, Singular Integral Equations, P. Noordhoff N.V.

[2] B. Epstein, Quarterly of Applied Mathematics, Vol, 6, No, 3, pp.301-

317 (Oct. 19U8).

[3] Z, Nehari, Conformal Mapping, McGraw-Hill,

-13-

(CLASSIFICATION)

Security Information

B"*.bliogr aphic al Control Sheet

1, Originating agency and /or monitoring agency ;

O.A.: Institute of Mathematical Sciences, Division of Electromagnetic

Research, New York University, New York City

M.A,: Mathematics Division, Office of Scientific Research

2, Originating agency and /or monitoring agency report number ;

O.A,: Research Report Noo BR. -7

M.A.: OSR-TN-5U-195

3, Title and classification of title ; Determination of Coefficients of Capacitance

of Regions Bounded by Collinear Slits and of Related Regions

(UNCLASSIFIED)

U. Personal author(s) ; Bernard Epstein

5. Date of report ; August, 19%

6. Pages; 13 + 1

7. Illustrative material : Figvires 1, 2, 3, k, S (drawings)

8. Prepared fer Contract No.(s) : AF-l8(600)367

9* Prepared for Project Code(s ) and/or No.(s) : No. R-35ii-10-l5

10. Security classification : UNCLASSIFIED

11. Distribution limitations ; none

12. Abstract :

A number of formulas are derived for the coefficients of capacitance of a

domain which consists of the entire plane minus any finite number of col-

linear slits. It is shown that any of a broad class of domains possessing

a certain symmetry property is confonnally eqxiivalent to such a 'slit-domain,'

and that the problem of determining the conf ormal correspondence is essen-

tially one of mapping simply-connected domains rather than multiply-con-

nected ones. As an illustration, the coefficients of capacitance of a Â»bi-

filar shielded cable' are computed.

v'c(!^'

Date Due

i

-1 PRINTE

J

D IN U. S. A.

/ - . . u

7 Epstein

jjeterrnination of coefficients

of capacitance of regions...

- TITLE

IIYTJ

3R-

7 Epstein

AUTHOR "

i^e.-erini nation of coef f iciei. ' .

o?'- capacitance of regions,.

DATE

DUE

BORROWERS NAME

ROOM

NUMBER

N.Y.U. Courant Institute of

Mathematical Sciences

251 Mercer St.

New York 12, N. Y.

Manufactured in the United States for New York University Press

by the University's Office of Publications and Printing

NEW YORK UNIVERSITY

â– pOURANT INSTITUTE - LIBRARY

1251 MÂ«rcer St. New York, N.Y. 10012

/0^Â». ^x T^. NEW YORK UNIVERSITY

^ I ^ lY ^ Institute of Mathematical Sciences

0-1

Wl

s Division of Electromagnetic Research

cccxx^

'^^crr yX+

RESEARCH REPORT No. BR-7

Determination of Coefficients of Capacitance of Regions

Bounded by Collinear Slits and of Related Regions

B. EPSTEIN

CONTRACT No. AF- 1 8 {600)-367

AUGUST 1954

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

RESEARCH REPORT No. BR-7

DETERMINATION OF COEFFICIENTS OF CAPACITANCE OF REGIONS

BOUNDED BY COLLINEAR SLITS AND OF RELATED REGIONS

by

Bernard Epstein

/j. Q^-d-^^IL^

Bernard Epstein ^

P^f^^EW YORK UNIVERSITY

. POURANT INSTITUTE - LIBRARY

Morris Kline 25( Mercer St. New York, N.Y 10012

Project Director

^i^

The research reported in this document has been made possible through

support and sponsorship extended by the Office of Scientific Research,

H. Q. Air Research and Development Command, U. S. A. F. , Baltimore,

Maryland, under Contract No. AF 18(600)-367. It is published for tech-

nical information only, and does not necessarily represent recommenda-

tions or conclusions of the sponsoring agency.

New York, 1954

C2L

Abstract

A number of formulas are derived for the coefficients of capacitance of

a domain which consists of the entire plane minus any finite number of col-

linear slits. It is shown that any of a broad class of domains possessing

a certain symmetry property is conformally equivalent to such a 'slit -domain',

and that the problem of determining the conformal correspondence is essential-

ly one of mapping simply -connected domains rather than multiply-connected ones.

As an illustration, the coefficients of capacitance of a 'bi-filar shielded

cable' are computed.

Table of Contents

Page

1, Introduction 1

2, Regions Bounded by Collinear Slits 2

3Â» Extension to More General Regions 6

U, An Illustrative Example 8

References 12

- 1 -

1, Introduction

In the study of electrostatic field problems the principal objective

usually is to determine the potentisil and its gradient (the field strength)

throughout a given domain boimded by a system of conductors. Frequently,

however, it is necessary only to determine certain constants of capacitance.

In this paper vre consider the latter problem for a certain class of plane

domains.

Let n be a domain consisting of the entire x-y plane with any finite

number of slits along a single line. Several formulas for the coefficients

of capacitance of such a domain are derived, two of which appear to be well

suited for nu'nerical computations. One of these formulas is based on the ex-

plicit representation of the potential as the i^eal part of an analytic function I- J*' - '

while the other formula has the feature of requiring a knowledge of the po-

tential only on the line containing the boundary components of Dj it does

not involve any derivatives of the potential, A convenient method for de-

termining the field along the line containing the boundary components has

[2]

been givÂ«n by the author in a previous paper, -^

Since the coefficients of capacitance of a domain are invariant under

conformal mapping, the formulas which are derived may be employed to compute

the coefficients of any domain which can be conformally mapped upon a domain

D of the type described above. This procedure can be applied to a certain

class of domains which are of practical interest. In these cases the mapping

problem involves essentially only simply-connected domains rather than multi-

ply-connected domains. One particular case of interest, the 'bi-filar shield-

ed cable' is considered in some detail, and as an illustration of the pro-

cedure, the coefficients of capacitance are evaluated numerically for one

such domain.

- 2 -

2, Regions Boxinded by Collinear Slits

We consider here the problem of determining effectively the so-called

coefficients of capacitance of a domain D consisting of the extended (x,y)-

plane with a finite number, m, of collinear slits, cut along what may be assumed

to be the x-axisÂ»

We recall the definition of these coefficients:

/ Su.

(2.1) Pik^-cT aJT^s, (i, k = l,2, - ,m),

'^k

where u., the harmonic measure of the i-th boundarj'^ component, is the harmonic

function whose boundary values are unity on the i-th component and zero on all

other components (the existence and uniqueness of this harmonic function is

well known from the theory of the Dirichlet problem); C, is any curve, described

in the positive sense, surrounding only the k-th component; and â€” indicates

differentiation in the direction of the outward normal. As follows immediately

from the C auchy-Riemann equations, the p., may be defined alternatively as the

increment in the harm.onic f\incticn v. conjugate to u. which results when C, is

described cnce in the negative sense.

We also recall several important properties of these coefficients of cap-

acitance:

i = l,2,...,m;

(2.2a)

Pik ^ Pki 5

(2.2b)

r Pik - 0^

k=l

(2.2c)

Pii > ;

(2.2d)

Pik < ^^

i / k.

The p., are, of course, conformal invariants. Hence we may assume that D con-

sists of the entire plane minus a finite niomber of slits lying on the x-axis,

one of which extends to infinity in both directions (see Fig. 1); this con-

figuration can always be realized by a suitable inversion. It will be seen

- 3 -

that this assxunption eliminates the possibility of any convergence difficulties.

For brevity such domains will be called slit-domains. We number the finite slits

1, 2, â€¢â€¢â€¢, m-1 from left to right; and the infir.ite slit is the m-th.

y

Figure 1

-CD

Or

^2 ^3

m-i

a b

m m

+00

We proceed to derive various formulas for the quantities p., which may

prove useful for numerical computations. First we sha32 employ the definition

(2.1); later we shall derive a formula based on the alternative definition

given following (2.1),

Taking into account (2,2a) and (2.2b) we see that it suffices to determine

the p., with i < m, k < m. Let f . (x) = u.(x,0), so that, in particular,

f . (x) = for xb. By the Poisson formula we have, for any point

off the X-axis:

(2.3)

m f. (f)d4

I 1 / m f. (f

u (x,y) Â» M / ^â€”

T-T

a^ (4-x) +y

We take as the curve C, a rectangle with vertical sides passing through the

gaps (a, ,b, ) and (a, -i *ti, ,), It is easily found from (2.3), by differentiating

under the integral sign, that for any value of x:

^^i I 2A

dTl- ~ ' ^ " maxda^Mb^l)

(2.U)

l^^il 2A^

aT"! - ~3 '

' ' ny

Tiy

(one uses, of course, the fact that |f . (4) | 5 !)â€¢ Hence as the horizontal sides

recede to infinity the contribution to the integral (2.1) from these sides

vanishes, and therefore (2.1) may be written as follows:

(2.5)

ik i,k ~ i,k+l *

where

(2.6)

- U -

\i 'jC ^ dy, a. < X < b.

(of course p. -I. -I.,).

^ ^im i,m 1,1

Ws wish to rewrite I. .in such a form as to involve only the values of

i,j

au.

u. on the x-axis, i.e., the values f.(x). If one expresses ^ â€” by differ-

1 1 ox

entiating (2.3), inserts this expression into (2.6), and interchanges the order

of integration, one obtains the following expression for I. .:

(2.7)

P /m f (Â£)d4

I. , - - P / -ip , a. < X < b..

Here the symbol P denotes the Cauchy principal value of the integral.

A second expression for I. .is obtained by the follovring artifice:

Since the right side of (2.6) gives the same value for any choice of x in the

j-th gap, we integrate both sides of (2.6) over this gap. Then we obtain

'V^j>^i..i

b . .oo 3â€ž

'a . '^-00

aT- ^y^-

Interchanging the order of integration [this is easily justified with the aid

of the first inequality in (2.U)J we obtain

.oo

Â«oo h

(2.8) (b -a )I

J J i> J

^-00 a .

BU^

aF

CO

n 1^-00

|y|

m

Ih

< u(b ,y)-u(a.,y)[. dy

^(4) r ^-^ ^-j 1

^ K4-bJ V (4-aJ V J

d4 ydy.

Now we interchange the order of integration once more, and obtain the

following expression for I. .:

m

(2.9)

^i,j i^(^r^y^

f^U) In

4-a,

4-b

d4.

A third expression for I. .is obtained as follows. Since f.(4) is con-

1, J 1

tinuous and f.'(4) and In

- 5 -

^-^J

are absolutely integrable, an integration by

parts enables us to rewrite (2.9) in the form:

b

(2-lÂ°> hS iT^ / "q(f.)[(4-aj)lnK-a.|-(*,-bj)l.lS.b^|]dÂ£i

the integrated term vanishes since ^^^(^1) = -^i^^m^ " ^'

Equations (2.7), (2.9), (2.10), together with (2.5), give three formulas

for the coefficients p., which employ only the values of u. on the x-axis.

These equations were all derived using the definition (2,1) of p , . A fourth

formula is obtained by using the alternative definition, as follows. It is

shown in [l] , Â§ 91 that the analytic function v^{z) = u^+i v^ must satisfy

the condition

P,(z)

(2.11) wj(z) ^

i^ ' /^n

/. TF (z-a )(z-b )

where P.(z) is a polynomial, of degree not exceeding m-2, with ra-1 real co-

efficients which are uniquely determined by the conditions :

/j r +1, for j - i,

(2.12) / w.'(z)dz = { -1, for j Â« i+1,

â– 4. ^ 0, for ;j ?< i, j / i+1

(It is shown in reference [l] that of the m conditions (2.12) only m-1 are

independent.) Now employing the alternative definition of p , and taking

for C, the doubly-counted k'th interval, one easily obtains from (2.11) the

k

result

A+1 P.(C)dC

(2.13) Pik-^Vn

m *

/TT (4-a,)(?-b )

where the ambiguity in sign is most easily resolved with the aid of (2,2c)

and (2. 2d).

â€¢* It must be remembered that the denominator of w^(z) changes sign on

successive intervals (a., b.).

- 6 -

Insofar as numerical computations are concerned, it would appear that

formulas (2.9) and (2.13) are especially suitable. While (2.13) has an ad-

vantage over (2.9) in that integration is necessary only over a single interval,

it involves the solution of the system (2.12) for the coefficients of the poly-

nomial P.(z) and this may become laborious for large values of m. In this case

it might prove preferable to employ (2,9), obtaining the fvmction Â£^{x) to the

desired degree of approximation by the method given in [2^, Â§ 6.

3, Extension to More General Regions

Since the coefficients of capacitance are conformal invariants, they may

be determined by the method described above for any domain which can be mapped

confomially onto a slit-domain, A simple example of such a domain is the entire

(x,y)-plane slit along a finite number of arcs of a circle (cf, [l] , Â§ 92), By

a suitable linear transformation we can map the circumference of this circle

onto the x-axis, thus obtaining a slit-domain.

Another domain to which this method applies is a domain bounded externally

by one circle and internally by two others. By a suitable linear transformation

such a region may always be mapped into a domain with the center of all three

circles collinear (cf. Fig. 2), y

Figure 2

- 7 -

Now suppose that the upper half of this domain, which is simply-connected,

is mapped conforraally by an analytic function Y, ' f(z) upon the upper half of

the ^-plane. Then the upper halves of the three circles are mapped into seg-

ments of the real axis of the )^-plane, and the three segments of the axis of

s yi i u t i etry (AB, CD, EF) are mapped into the remainder of the real axis. By the

Schwarz reflection principle, the entire domain of FigÂ» 2 is then conformally

mapped onto the ^-plane with three slits along the real axis, i.e., upon a slit-

domain, A particular case of practical interest is the 'bi-filar shielded con-

ductor'. This case will be discussed in some detail in Section U,

Closely related to the configuration of Fig. 2 is that of the plane with

any finite number of circular apertui^s whose centers all lie on one line

(cf.Fig.3). Each half of this region is simply-connected. If, as before, we can make

O

^ i-^v^

1-2 ya.8

of the real axis. Thus the domain of Fig. h has been approximately mapped

into a slit-domain, and the formulas given there may be employed to determine

the p^j^.

- 11 -

To give a numerical illustration, let a - pQ , P â– r. Then p Â«â– ^, and,

since the infinity products (U.U) and (i;.5) converge rapidly, one obtains very

easily

l.f(iliJ^ ,p)

(U.6) ^^ - 0.212, ^ = U.726, "-^ >g - 8.628 .

1-2 y^

In order to work with more convenient numbers, we employ the transformation

(U.7) z - -^ z,

thus obtaining in the z-plane a slit-domain characterized by the following

numbers (cf. Fig. 1):

(U.8) a^-= -8.1, b^= -U.5, 3^= -0.2, b2= 0.2, a^= U.S, b^- 0.1.

On account of the low connectivity of the domain (m=3) it was decided to era-

ploy formula (2.13) to evaluate the p., . The coefficients ppp and Pp-, (the

numbering of the boundary components is given in Fig. U and Fig. 5, and is

in accordance with the numbering given preceding Fig. 1) were determined as

follows; The polynomial Pp(z), which according to the discussion of Â§2 is

2

of first degree, was replaced in (2,12) by the quadratic polynomial az +bz+c

with unknown coefficients a, b, c. The integrals appearing in (2,12) were

computed by interpolation (the Tchebyshev five-point formula was used) and

the resulting system of linear equations was solved for the coefficients a,

b, c, with the following results:

(1+.9) a = -9.3 X 10"^, b - -1.962, c - -11.608.

(The radical appearing in (2,12) was taken positive on the first and third

intervals and negative on the second interval.) The extremely small value

obtained for a, whose exact value must be zero, ser*ves as an excellent check

on the accuracy of the computations. The expression for Pp(z) thus obtained

- 12 -

was then employed in (2.13), with the following results:

(li.lO) P2^ - -2.153, P22 - +3.820,

3y symmetry, p^, â– Poo> ^^"^ ^'^^ ^^^ remaining six coefficients are obtained

with the aid of (2.2a) and (2.2b). Thus, the following table of values of

the p., was obtained.

2

3

+3.320

-2.153

-1,667

-2 ,153

+3.320

-1.667

-1.667

-1,667

+3.33U

As a check, p-p was also computed by means of (2.5) and (2.10), and excellent

agreement was obtained.

Acknowledgment

We wish to thank Mr, Charles Kahane, who so capably carried out the

computations connected with the illustrative example of Section U,

References

[1] N, I, Muskhelishvili, Singular Integral Equations, P. Noordhoff N.V.

[2] B. Epstein, Quarterly of Applied Mathematics, Vol, 6, No, 3, pp.301-

317 (Oct. 19U8).

[3] Z, Nehari, Conformal Mapping, McGraw-Hill,

-13-

(CLASSIFICATION)

Security Information

B"*.bliogr aphic al Control Sheet

1, Originating agency and /or monitoring agency ;

O.A.: Institute of Mathematical Sciences, Division of Electromagnetic

Research, New York University, New York City

M.A,: Mathematics Division, Office of Scientific Research

2, Originating agency and /or monitoring agency report number ;

O.A,: Research Report Noo BR. -7

M.A.: OSR-TN-5U-195

3, Title and classification of title ; Determination of Coefficients of Capacitance

of Regions Bounded by Collinear Slits and of Related Regions

(UNCLASSIFIED)

U. Personal author(s) ; Bernard Epstein

5. Date of report ; August, 19%

6. Pages; 13 + 1

7. Illustrative material : Figvires 1, 2, 3, k, S (drawings)

8. Prepared fer Contract No.(s) : AF-l8(600)367

9* Prepared for Project Code(s ) and/or No.(s) : No. R-35ii-10-l5

10. Security classification : UNCLASSIFIED

11. Distribution limitations ; none

12. Abstract :

A number of formulas are derived for the coefficients of capacitance of a

domain which consists of the entire plane minus any finite number of col-

linear slits. It is shown that any of a broad class of domains possessing

a certain symmetry property is confonnally eqxiivalent to such a 'slit-domain,'

and that the problem of determining the conf ormal correspondence is essen-

tially one of mapping simply-connected domains rather than multiply-con-

nected ones. As an illustration, the coefficients of capacitance of a Â»bi-

filar shielded cable' are computed.

v'c(!^'

Date Due

i

-1 PRINTE

J

D IN U. S. A.

/ - . . u

7 Epstein

jjeterrnination of coefficients

of capacitance of regions...

- TITLE

IIYTJ

3R-

7 Epstein

AUTHOR "

i^e.-erini nation of coef f iciei. ' .

o?'- capacitance of regions,.

DATE

DUE

BORROWERS NAME

ROOM

NUMBER

N.Y.U. Courant Institute of

Mathematical Sciences

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New York 12, N. Y.

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