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Homework 3 Math 587

1. (Bender & Orszag 6.18(b)) Consider Z ∞ cos(xt) dt. t 1 (a) For which limit(s), x → 0+ and/or x → ∞, is integration by parts useful? (b) Find the leading order behavior for x → 0+ (hint: substitute s = xt. What is the behavior near s = 0?) 2. (Bender & Orszag 6.28(c)) Find leading behavior for x → ∞ of Z

π/4

√ 2 tan t e−xt dt.

0

3. Find leading behavior for x → ∞ of Z ∞ exp(x[t2 − t4 ])dt −∞

4. Find the leading order behavior as x → ∞ for Z x t ee dt. 0

Hint: try a substitution to bring the integral into Laplace form.

5. Consider an integral like Z

∞

f (t) exp(ixψ(t))dt

0

which has a single stationary phase point at t = 0. Suppose that f ∼ t and ψ(t) ∼ at3 as t → 0. (a) Evaluate Z ∞

t exp(ixt3 )dt.

−∞

(Hint: choose complex contours t = seiφ so that the exponential becomes real) (b) Is it justified to find the leading order behavior for x → ∞ by simply replacing f and ψ with their leading order terms? (Note that for f ∼ t and ψ ∼ t2 , this is not the case; integration by parts was needed to bring the integral into stationary phase form). (c) (Similar to Bender & Orszag 6.56(e)) Find leading order behavior as x → ∞ for Z 1

sin[x(t − sin t)] sinh t dt

−1

6. Using the method of steepest descents, find the leading order approximation as x → ∞ for Z ∞ cos(xt) exp(−1/t)dt.

I(x) =

0

(Hint: this is similar to the “moving maximum” LaplaceR integrals where one needs a change of variables to bring this into the form exp(λf (t))dt where λ → ∞.) 7. Consider the integral

Z

C

exp[x(sinh t − t)]dt

where C is the contour t = s2 + 2i arctan s, −∞ < s < ∞. (a) Plot some of the steepest descent contours. (b) Determine the leading behavior as x → ∞...