14—Complex Variables 36914.35 Evaluate the integral of problem14.33another 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 infinite series. Will section1.2save you any work? Ans: Still 2 πλ/(λ^2 −1)^3 /^2
14.36 Evaluate
∫∞0dxcosαx^2 and
∫∞
0dxsinαx^2 by considering
∫∞
0dxeiαx
2Push the contour of integration toward the 45 ◦line. Ans:^12
√
π/ 2 α
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?
14.38 Derive ∫∞
0
dx
1
a^3 +x^3
= 2π
√
3 / 9 a^2
14.39 Go back to problem3.45and find the branch points of the inverse sine function.
14.40 What is the Laurent series expansion of 1 /(1 +z^2 )for small|z|? Again, for large|z|? What is
the domain of convergence in each case?
14.41 Examine the power series
∑∞
0 z
n!. What is its radius of convergence? What is its behavior asyou move out from the origin along a radius at a rational angle? That is,z =reiπp/qforpandq
integers. This result is called a natural boundary.
14.42Evaluate the integral Eq. (14.10) for the casek < 0. Combine this with the result in Eq. (14.15)
and determine if the overall function is even or odd ink(or neither).
14.43 At the end of section14.1several differentiation formulas are mentioned. Derive them.
14.44 Look at the criteria for the annulus of convergence of a Laurent series, Eqs. (14.7) and (14.8),
and write down an example of a Laurent series that converges nowhere.
14.45 Verify the integral of example 8 using elementary methods. It will probably take at least three
lines to do.
14.46 What is the power series representation forf(z) =
√
zabout the point1 +i? What is the
radius of convergence of this series? In the Riemann surface for this function as described in section
14.7, show the disk of convergence.