- #1

Hoplite

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I have the question,

[tex] \int_{-\pi/2}^{\frac{\pi}{2}} e^{-ilk}cos^n kdk[/tex]

It says, "Set t=ik". So,

[tex]-i\int_{-i\pi/2}^{i\pi/2}e^{-lt} cosh^n tdt[/tex]

But then it says, "Use the method of steepest descent to show that as n [tex]\rightarrow \infty[/tex] with r = l/n."

I'm supposed to get:

[tex]\sim \sqrt{\frac{2\pi}{n(1-r^2)} }exp(-\frac{1}{2}n[r\log{\frac{1+r}{1-r}}+log(1-r^2)])[/tex]

If the equation were of the form, [tex]\int e^{ilP(t)}Q(t)dt[/tex], I know how to use the method of steepest descent. I'd find a point z where P'(t)=0 and expand P(t) around that point using a Taylor series expansion getting, P(t)=P(z)+0.5P''(z)(t-z)^2, and then I'd replace t with z+ix and it would all come out from there. But I have no idea how to use the method of steepest descent when P(t)=t and i has been removed from the exponential.

[tex] \int_{-\pi/2}^{\frac{\pi}{2}} e^{-ilk}cos^n kdk[/tex]

It says, "Set t=ik". So,

[tex]-i\int_{-i\pi/2}^{i\pi/2}e^{-lt} cosh^n tdt[/tex]

But then it says, "Use the method of steepest descent to show that as n [tex]\rightarrow \infty[/tex] with r = l/n."

I'm supposed to get:

[tex]\sim \sqrt{\frac{2\pi}{n(1-r^2)} }exp(-\frac{1}{2}n[r\log{\frac{1+r}{1-r}}+log(1-r^2)])[/tex]

If the equation were of the form, [tex]\int e^{ilP(t)}Q(t)dt[/tex], I know how to use the method of steepest descent. I'd find a point z where P'(t)=0 and expand P(t) around that point using a Taylor series expansion getting, P(t)=P(z)+0.5P''(z)(t-z)^2, and then I'd replace t with z+ix and it would all come out from there. But I have no idea how to use the method of steepest descent when P(t)=t and i has been removed from the exponential.

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