Mathematical Tools for Physics

14—Complex Variables 449

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. (2.14). Ans:π 2 nCn

/

22 n−^1 =π(2n−1)!!/(2n)!!

14.35 Evaluate the integral of problem 33 another way. Assumeλis large and expand the integrand in a power
series in 1 /λ. Use the result of the preceding problem to evaluate the individual terms and then sum the resulting
series. Ans: Still 2 πλ/(λ^2 −1)^3 /^2

14.36 Evaluate
∫∞

0

dxcosx^2 and

∫∞

0

dxsinx^2 by considering

∫∞

0

dxeix

2

Push the contour of integration toward the 45 ◦line. Ans:^12

√

π/ 2

14.37

f(z) =

1

z(z−1)(z−2)

−

1

z^2 (z−1)^2 (z−2)^2

What is

∫

Cdz f(z)about the circlex

(^2) +y (^2) = 9?