MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 10 pps

12.14 Equipotentials and electric field lines in the w-plane parallel plates held at potentials Vl and .V2, where the electric field lines are given by the family of lines u = z2 - y2 =

Fig 12.12 Conformal mapping

12.4.1 Conformal Mappings

To see an interesting property of analytic functions we differentiate

a t zo, where the modulus and the arguments of the derivative are given as

I $ I and a, respectively We now use polar coordinates to write the modulus

20

lim

At-0

and the argument (Fig 12.12) as

a = lim arg[Azu] - lim arg[Az]

(12.116) ( 12.1 17)

(12.118) (12.119) Since this function, f ( z ) , maps a curve c, in the z-plane into another curve

c, in the w-plane, from the arguments [& (12.119)] it is seen that if the slope of c, at is 00, then the slope of cw at wo is a+& For a pair of curves intersecting a t a the angle between their tangents in the w- and z-planes will

be equal, that is,

4'2 - 41 = (02 + a ) - (0, + a ) , ( 12.120)

Hence analytic functions preserve angles between the curves they map (Fig 12.12) For this reason they are also called conformal mappings or transfor- mations

314 COMPLEX VARIABLES AND FUNCTIONS

fig 12.13 T w o plates with hyperbolic cross sections

12.4.2 Electrostatics and Conformal Mappings

Conformal mappings are very useful in electrostatic and laminar (irrotational) flow problems, where the Laplace equation must be solved Even though the method is restricted to cases with one translational symmetry, it allows one

t o solve analytically some complex boundary value problems

Example 12.10 Conformal mappings and electrostatics: Let us con- sider two conductors held at potentials Vi and V2 with hyperbolic cross sections

x2 - 9’ = c1 and x2 - y2 = c2 (12.122)

We want to find the equipotentials and the electric field lines In the complex z-plane the problem can be shown as in Figure 12.13 We use the conformal mapping

fig 12.14 Equipotentials and electric field lines in the w-plane

parallel plates held at potentials Vl and V2, where the electric field lines are given by the family of lines

(u =) z2 - y2 = cz (12.129)

In three dimensions, to find the equipotential surfaces these curves must

be extended along the direction of the normal to the plane of the paper

Example 12.11 Electrostatics and conformal mappings: We now find

the equipotentials and the electric field lines inside two conductors with semicircular cross sections separated by an insulator and held at poten- tials +V, and -V& respectively (Fig 12.15) The equation of a circle

316 COMPLEX VARIABLES AND FUNCTIONS

+ z-plane

Fig 12.15 T w o conductors with semicircular cross sections

in the z-plane is given as

4

U

Fig 12.16 Two semicircular conductors in the w-plane

From the limits

318 COMPLEX VARIABLES AND FUNCTIONS

found by extending these curves in that direction The solution to this problem has been found rather easily and in closed form Compare this with the separation of variables method, where the solution is given

in terms of the Legendre polynomials as an infinite series However, applications of conformal mapping are limited to problems with one translational symmetry, where the problem can be reduced t o two di- mensions Even though there are tables of conformal mappings, it is not always easy as in this case to find an analytic expression for the needed mapping

This equation alone is not sufficient to determine the velocity field *(?",t)

If the flow is irrotational, it will also satisfy

? X T = O , (12.147) thus the two equations

and

Fig 12.17 Flow around a wall of height h

completely specify the kinematics of laminar, frictionless flow of incompress- ible fluids These equations are also the expressions of linear and angular momentum conservations for the fluid elements Fluid elements in laminar flow follow streamlines, where the velocity g(?",t) a t a given point is tan- gent to the streamline a t that point

Equations (12.148) and (12.149) are the sarme as Maxwell's equations in electrostatics Following the definition of electrostatic potential, we use Equa- tion (12.149) to define a velocity potential as

(12.150) + + 21 ( T ,t) = T k q 7 , t )

Substituting this into Equation (12.148) we obtain the Laplace equation

V2@(7, t ) = 0 (12.151)

We should note that even though ẳ", t ) is known as the velocity potential

it is very different from the electrostatic potential

Example 12.12 Flow around an obstacle of height h: Let us consider

laminar flow around an infinitely long and thin obstacle of height h Since

the problem has translational symmetry, we can show it in two dimen- sions as in Figure 12.17, where we search for a solution of the Laplace equation in the region R

Even though the velocity potential satisfies the Laplace equation like the electrostatic potential, we have to be careful with the boundary conditions In electrostatics, electric field lines are perpendicular to the equipotentials; hence the test particles can only move perpendicular to the conducting surfaces In the laminar flow case, motion perpendicular

320 COMPLEX VARIABLES AND FUNCTIONS

to the surfaces is not allowed because fluid elements follow the contours

of the bounding surfaces For points far away from the obstacle, we take the flow lines as parallel to the z-axis As we approach the obstacle, the

flow lines follow the contours of the surface For points away from the

obstacle, we set

We now look for a transformation that maps the region R in the z-plane

to the upper half of the w-plane Naturally, the lower boundary of the region R in Figure 12.17 will be mapped to the real axis of the w-plane

We now construct this transformation in three steps: We first use

to map the region R to the entire wl-plane Here the obstacle is between

0 and -h2 As our second step, we translate the obstacle to the interval between 0 and h2 by

~2 = z2 + h2 (12.154) Finally we map the w2-plane to the upper half of the w-plane by

wz= w , + h2

I

fig 12.18 Transition from the z-plane to the w-plane

322 COMPLEX VARIABLES AND FUNCTIONS

h

T’

4 z-plane

fig 12.20

upper half of the w-plane

Schwara-Christoffel transformation maps the inside of a polygon to the

12.4.4 Schwarz-Christoffel Transformations

We have seen that analytic transformations are also conformal mappings, which preserve angles We now introduce the Schwarz-Christoffel transfor- mations, where the transformation is not analytic at an isolated number of points Schwarz-Christoffel transformations map the inside of a polygon in

the z-plane to the upper half of the w-plane (Fig 12.20) To construct the Schwarz-Christoffel transformations let us consider the function

(12.156)

where A is complex, kl is real, and w1 is a point on the u-axis Comparing the arguments of both sides in Equation (12.156) we get

arg (2) = lim [arg Az - arg Awl

Aw-0

lim [arg Az - arg Awl =

As we move along the positive u-axis

lim arg Aw = arg [dw] = 0,

Aw-0

hence we can write

lim [arg Az] = arg[dz] =

(12.157)

(12.158)

(12.159)

For a constant A this means that the transforwation [Eq (l2.156)] maps the

parts of the u-axis; w < w1 and w > w1, to two line segments meeting at zo

in the z-plane Thus

A (w - W I ) - ~ ' (12.160) corresponds to one of the vertices of a polygon with the exterior angle k l ~ and located a t z1 For a polygon with n-vert.ices we can write the Schwarz- Christoffel transformation as

dz

- dw = A (W - ~ 1 ) (W ~ - ~w2)-IC2 ' (w - w,)-~" (12.161) Because the exterior angles of a polygon add up to 21r, powers ki should satisfy the condition

A polygon can be specified by giving the coordinates of its n vertices in the z-plane Along with the constraint [Eq (12.162)] this determines the 2 n f l of the parameters in the transformation This means that we have the freedom

to choose the locations of the three wi on the real axis of the w-plane

324 COMPLEX VARIABLES AND FUNCTIONS

fig 12.21 Region we map in Example (12.13)

Example 12.13 Schwarz-ChristofSel transformation: We now construct

a Schwarz-Christoffel transformation that maps the region shown in Fig- ure 12.21 to the upper half of the w-plane Such transformations are frequently needed in applications To construct the Schwarz-Christoffel transformation we define a polygon whose inside, in the limit as z3 +

-00, goes t o the desired region (Fig 12.22) Using the freedom in defining the Schwarz-Christoffel transformation we map the points z1 ,

2 2 , and z3 to the points

w3 + 00, w1=-1, w 2 = + 1 (12.163)

in the w-plane We now write the Schwarz-Christoffel transformation as

(12.164) Powers lcl, lcg, and lc3 are determined from the figure as f , f, and 1, respectively Note how the signs of lc; are chosen as plus because of the counterclockwise directions shown in Figure 12.22 Because the constant

c is still arbitrary, we define a new finite complex number A as

fig 12.22

the limit z3 -+ 00

The polygon whose interior goes to the desired region in Example 12.13 in

This can be integrated as

z = A cosh-' w + B , (12.168) where the constants A and B are found from the locations of the vertices, that is

(12.171)

Semi-infinite parallel plate capacitor: V - now calcu- late the fringe effects in a semi-infinite parallel plate capacitor Making use of the symmetry of the problem we can concentrate on the region shown in Figure 12.23 To find a Schwarz-Christoffel transformation that maps this region into the upper half of the w-plane we choose the points on the real w-axis as

1

I z4 z1 -+ -+ w4+ w1 + +a -m (12.172)

326 COMPLEX VARIABLES AND FUNCTIONS

, z-plane

Fig 12.23 Semi-infinite parallel plate capacitor

Since kz = -1 and Ic3 = 1, we can write

z = c [1w[ ei+ +In lzul+24] + D

Considering the limit in Figure 12.24 we can write

z,PPer - z p w e r - zd Using Equation (12.176) this becomes

fig 12.24 Limit of the point z3

thus determining the constant c as

328 COMPLEX VARIABLES AND FUNCTIONS

fig 12.25 w-Plane for the semi-infinite parallel plate capacitor

Fig 12.26 Z-Plane for the semi-infinite parallel plate capacitor

This gives us the parametric expression of the equipotentials in the z-

plane (Fig 12.27) as

sin (:T) + q d V

y = ?re

(12.186) (12.187) Similarly, the electric field lines in the ?-plane are written as

Transforming back to the z-plane, with the definitions

fig 12.27 Equipotentials for the semi-infinite parallel plate capacitor

330 COMPLEX VARIABLES AND FUNCTIONS

u(2, y) = sin z cosh y + x2 - y2 + 4xy

is a harmonic function and find its conjugate

12.3 Show that

u(z, y) = sin 2z/(cosh 2y + cos 2z)

can be the real part of an analytic function f ( z ) Find its imaginary part and express f ( z ) explicitly as a function of z

12.4 Using cylindrical coordinates and the method of separation of variables find the equipotentials and the electric field lines inside two conductors with semi-circular cross sections separated by an insulator and held at potentials +VO and -VO, respectively (Fig 12.15) Compare your result with Example 12.11 and show that the two methods agree

12.5 With aid of a computer program plot the equipotentials and the elec- tric field lines found in Example 12.14 for the semi-infinite parallel plate ca- paci t or

12.6 In a two-dimensional potential problem the surface ABCD is at poten- tial VO and the surface EFG is at potential zero Find the transformation (in differential form) that maps the region R into the upper half of the w-plane (Fig 12.28) Do not integrate but determine all the constants

12.7 Given the following twc+dimensional potential problem in Figure 12.29,

The surface ABC is held at potential VO and the surface DEF is at potential zero Find the transformation that maps the region R into upper half of the

w-plane Do not integrate but determine all the constants in the differential form of the transformation

12.8 Find the Riemann surface on which

J(z - 1)(z - 2)(z - 3)

4 2- plane

Fig 12.28 Two-dimensional equipotential problem

-m

Fig 12.29 Schwartz-Christoffel !.ransformation

8 Find the Riemann surface on which

d(z - l)(z - 2)( z - 3)

is single valued and analytic except a t z := 1,2,3

9 Find the singularities of

f( z ) = tanh L

10 Show that the transformation

332 COMPLEX VARIABLES AND FUNCTIONS

t

fig 12.30 Rectangular region surrounded by metallic plates

or

maps the 21 =const lines into circles in the z-plane

12.11 Use the transformation given in Problem 12.10 to find the equipoten-

tials and the electric field lines for the electrostatics problem of two infinite

parallel cylindrical conductors, each of radius R and separated by a distance

of d, and held at potentials +VO and -Vo, respectively

12.12 Consider the electrostatics problem for the rectangular region sur-

rounded by metallic plates as shown in Figure 12.30 The top plate is held

at potential VO, while the bottom and the right sides are grounded (V = 0)

The two plates are separated by an insulator Find the equipotentials and the

electric field lines and plot

12.14 Find the equipotentials and the electric field lines for a conducting

circular cylinder held at potential VO and parallel to a grounded infinite con-

ducting plane (Fig 12.32) Hint: Use the transformation z = a tanhiw/2

-

fig 12.31 %angular region

fig 12.32 Conducting circular cylinder parallel to infinite metallic plate

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13

COMPLEX INTEGRALS and

SERIES

In this chapter we first introduce the complex integral theorems Using ana- lytic continuation we discuss how these theorenis can be used to evaluate some frequently encountered definite integrals In conjunction with our discussion

of definite integrals, we also introduce the gamma and beta functions We also introduce complex series and discuss classification of singular points

13.1 COMPLEX INTEGRAL THEOREMS

I Cauchy-Goursat Theorem

Let C be a closed contour in a simply connected domain (Fig 13.1) If

a given function, f ( z ) , is analytic in and on this contour, then the integral

is true

Proof We write the function f (2) as

335

336 COMPLEX INTEGRALS AND SERIES

Using the Stokes theorem

we can write integral (13.3) as

(u + iv) ( d z + i d y )

= SS, (-& da: - ”> dY d s + SJ, (2 - $) ds, (13.5) where 5’ is an oriented surface bounded by the closed path C Because the Cauchy-Riemann conditions are satisfied in and on the closed path

C, the right-hand side of Equation (13.5) is zero, thus proving the the-

orem

11 Cauchy Integral Theorem

If f ( z ) is analytic in and on a closed path C in a simply connected domain

13.2) and if ~0 is a point inside the path C, then we can write the (Fig

integral

(13.6)