Mathematical Tools for Physics - Department of Physics - University

14—Complex Variables 368

whereCis a circle of radiusπnabout the origin. Ans:− 4 πin

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) = 2. 2796 , a modified Bessel function.

14.27 Evaluate

∫∞

0 dz/(a

(^4) +x (^4) ), and to check, compare it to the result of Eq. (14.15).
14.28 Show that ∫∞
0

dx

cosbx

a^2 +x^2

=

π

2 a

e−ab (a, b >0)

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

14.32 Evaluate ∫∞

0

dx

lnx

a^2 +x^2

(What happens if you consider(lnx)^2 ?) Ans:(πlna)/ 2 a

14.33 Evaluate(λ >1)by contour integration

∫ 2 π

0

dθ

(

λ+ sinθ

) 2

Ans: 2 πλ/(λ^2 −1)^3 /^2

14.34 Evaluate ∫π

0

dθsin^2 nθ

Recall Eq. (2.19). Ans:π 2 nCn

/

22 n−^1 =π(2n−1)!!/(2n)!!