Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

V 0 eiωt ̃

IR

L

L

C

C

A

B

DE

R

Figure 25.15 The inductor–capacitor–resistor network for exercise 25.1.

which can also be simplified, and gives

+

1

2

√

π(−x)^1 /^4

exp

[

i

(

2

3

(−x)^3 /^2 −

π

4

)]

.

Adding the two contributions and taking the real part of the sum, though this is not
necessary here because the sum is real anyway, we obtain

F(x)=

2

2

√

π(−x)^1 /^4

cos

(

2

3

(−x)^3 /^2 −

π

4

)

=

1

√

π(−x)^1 /^4

sin

(

2

3

(−x)^3 /^2 +

π

4

)

,

in agreement with the asymptotic form given in (25.53).

25.9 Exercises

25.1 In the method of complex impedances for a.c. circuits, an inductanceLis
represented by a complex impedanceZL=iωLand a capacitanceCbyZC=
1 /(iωC). Kirchhoff’s circuit laws,
∑

i

Ii= 0 at a node and

∑

i

ZiIi=

∑

j

Vjaround any closed loop,

are then applied as if the circuit were a d.c. one.
Apply this method to the a.c. bridge connected as in figure 25.15 to show
that if the resistanceRis chosen asR=(L/C)^1 /^2 then the amplitude of the
current,IR, through it is independent of the angular frequencyωof the applied
a.c. voltageV 0 eiωt.
Determine how the phase ofIR, relative to that of the voltage source, varies
with the angular frequencyω.
25.2 A long straight fence made of conducting wire mesh separates two fields and
stands one metre high. Sometimes, on fine days, there is a vertical electric field
over flat open countryside. Well away from the fence the strength of the field is
E 0. By considering the effect of the transformationw=(1−z^2 )^1 /^2 on the real and