Using the method of steepest descent

• Hoplite
In summary, the conversation discusses the use of the method of steepest descent to solve a specific integral with a given equation. The first step is to set t=ik, then use the method of steepest descent as n approaches infinity with r=l/n. The provided equation and a specific method for using the method of steepest descent are mentioned, but there is uncertainty about how to use the method when the exponential term has been modified. A link is also provided for further assistance.
Hoplite
I have the question,
$$\int_{-\pi/2}^{\frac{\pi}{2}} e^{-ilk}cos^n kdk$$
It says, "Set t=ik". So,
$$-i\int_{-i\pi/2}^{i\pi/2}e^{-lt} cosh^n tdt$$
But then it says, "Use the method of steepest descent to show that as n $$\rightarrow \infty$$ with r = l/n."
I'm supposed to get:
$$\sim \sqrt{\frac{2\pi}{n(1-r^2)} }exp(-\frac{1}{2}n[r\log{\frac{1+r}{1-r}}+log(1-r^2)])$$
If the equation were of the form, $$\int e^{ilP(t)}Q(t)dt$$, 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.

Last edited:
HINT: Write $\cosh ^n x = e^{\ln \cosh^n x}$

Thanks, Tide. That question was killing me.

1. What is the method of steepest descent?

The method of steepest descent is an optimization technique used to find the minimum value of a function. It involves taking small steps in the direction of the steepest slope of the function in order to reach the minimum value.

2. When is the method of steepest descent used?

The method of steepest descent is commonly used in numerical analysis and optimization problems. It is particularly useful in cases where the function is difficult to differentiate, but can still be evaluated at different points.

3. How does the method of steepest descent work?

The method of steepest descent works by starting at an initial point and taking small steps in the direction of the steepest slope of the function. This process is repeated until the function reaches a minimum value or until a stopping criterion is met.

4. What are the advantages of using the method of steepest descent?

The method of steepest descent is a relatively simple and efficient optimization technique. It can be used on a wide range of functions and does not require the function to be differentiable. It also converges quickly to a minimum value, making it a popular choice for optimization problems.

5. Are there any limitations to using the method of steepest descent?

One limitation of the method of steepest descent is that it may get stuck in a local minimum instead of reaching the global minimum. It also requires the function to be continuous and have a well-defined gradient. In some cases, it may also require a large number of iterations to reach the minimum value.

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