Mathematical Tools for Physics

14—Complex Variables 448

14.25 Evaluate the residues of these functions at their singularities. a,b, andcare distinct. Six answers: you
should be able to do five of them in your head.

(a)

1

(z−a)(z−b)(z−c)

(b)

1

(z−a)(z−b)^2

(c)

1

(z−a)^3

14.26 Evaluate the residue at the origin for the function

1

z

ez+

(^1) z
The result will be an infinite series, though if you want to express the answer in terms of a standard function you
will have to hunt. Ans:I 0 (2), a modified Bessel function.
14.27 Evaluate

∫∞

0 dz/(a

(^4) +x (^4) ), and as a check, compare it to the result of example 4, Eq. ( 12 ).
14.28 Evaluate ∫∞
0
dx
cosbx
a^2 +x^2
14.29 Evaluate (areal)
∫∞
−∞
dx
sin^2 ax
x^2
Ans:|a|π
14.30 Evaluate ∫
∞
−∞
dx
sin^2 bx
x(a^2 +x^2 )
14.31 Evaluate the integral

∫∞

0 dx

√

x/(a+x)^2. Use the ideas of example 8, but without the logarithm. (a > 0 )
Ans:π/ 2

√

a