Find the path of steepest descent

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In summary: This conversation discusses a function F(a) defined by F(a)=-ie^{-i\pi a/2}\int_{\pi/2-i\infty}^{\pi/2+i\infty}e^{ia(e^{iz}+z)}dz. The integral is shown to be convergent for real and positive values of a, and the saddle point(s) of the exponent are found by setting its derivative to zero. However, the path of steepest descent is unclear and further clarification is needed.
  • #1
karlzr
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Homework Statement


A function F(a) is defined by [tex]F(a)=-ie^{-i\pi a/2}\int_{\pi/2-i\infty}^{\pi/2+i\infty}e^{ia(e^{iz}+z)}dz[/tex]
where the integration is along the vertical line ([itex]Re(z)=\pi/2[/itex]).
(a) Show that the integral is convergent for real and positive values of a.
(b) Find the saddle point(s) of the exponent.
(c) Find the path of steepest descent.2. The attempt at a solution
I can handle the first two questions. However, I don't quite understand the last one. How to find the descent along a particular path? What does the last question ask about exactly?
 
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(a) To show that the integral is convergent for real and positive values of a, we need to show that the integrand is bounded. We can do this by using the absolute value of the integrand:|e^{ia(e^{iz}+z)}| = e^{-aIm(e^{iz}+z)}Now, since Re(z)=\pi/2, we have that Im(e^{iz}+z) \geq 0, so e^{-aIm(e^{iz}+z)} \leq 1. Since the integrand is bounded, the integral is convergent for real and positive values of a.(b) The saddle point(s) of the exponent can be found by setting its derivative to zero, i.e.\frac{d}{dz}ia(e^{iz}+z)=0This yields the equatione^{iz}+z=0which has the solution z_0=\pi/2+i\ln(-1).(c) I don't understand how to find the path of steepest descent.
 

Related to Find the path of steepest descent

What is the concept of "Find the path of steepest descent" in science?

The concept of "Find the path of steepest descent" is a method used in optimization problems to find the direction of the steepest decrease in a given function. This method is often used in physics, engineering, and other fields where minimizing a quantity is desired.

How is the path of steepest descent determined?

The path of steepest descent is determined by finding the gradient or slope of the function at a given point. This gradient vector points in the direction of the steepest decrease, and the path can be followed by taking small steps in this direction.

What is the significance of finding the path of steepest descent?

Finding the path of steepest descent is important because it allows us to efficiently minimize a given quantity. This method is often used in optimization problems, such as finding the minimum of a cost function in machine learning algorithms.

Are there any limitations to using the path of steepest descent?

Yes, there are limitations to using the path of steepest descent. This method may not always lead to the global minimum, and it can be sensitive to the initial starting point. Additionally, it may take longer to converge compared to other optimization methods.

How is the path of steepest descent related to gradient descent?

The path of steepest descent and gradient descent are closely related. In fact, gradient descent is a specific type of path of steepest descent that uses a fixed step size to update the parameters in each iteration. However, other variations of the path of steepest descent may use different methods for determining the step size, such as backtracking line search.

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