Finding the zeros of a complex function in PyLab

In summary, finding the zeros of a complex function in PyLab involves using the roots function to compute the roots of the function and then plotting them on a complex plane. This process can be useful for analyzing the behavior of the function and determining its critical points. Additionally, the use of the linspace function can help to visualize the zeros more accurately. By understanding the concept of complex numbers and using PyLab's tools, finding the zeros of a complex function becomes a straightforward task.
  • #1
PeteyCoco
38
1
I have this characteristic equation for the wave number eigenvalues [itex] k_n [/itex] of a homogeneous infinite cylinder of radius R:

[itex] D_{m} = (k_n R) = 0, [/itex]

where

[itex] D_m (z) = n_r J'_m(n_r z)H_m(z) - J_m(n_r z)H'_m(z) [/itex]

and [itex] n_r [/itex] is the refractive index of the cylinder, the bessel and hankel functions are both of the first kind, and z is a complex argument. I'm not sure how I can solve for the zeros of this. I've been using PyLab, but haven't found any clues as to what I should do.

The article I'm working from is: http://arxiv.org/abs/1302.0245
These are equations (18) and (19) from the article

I'm new to this, so I may be trying something ridiculous.
 
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  • #2
An even simpler statement of my problem: How can I find the zeros of a complex function in PyLab? Most sources I've found haven't helped.
 

1. What are complex functions in PyLab?

Complex functions in PyLab are mathematical functions that involve complex numbers, which are numbers that contain both a real and an imaginary component. They are typically represented in the form f(z) = u(x,y) + iv(x,y), where u and v are real-valued functions and i is the imaginary unit.

2. Why do we need to find the zeros of a complex function?

Finding the zeros of a complex function is important because it helps us understand the behavior and properties of the function. The zeros, also known as roots, are the values of z for which f(z) = 0, and they can provide insights into the behavior of the function near those points.

3. How do I find the zeros of a complex function in PyLab?

To find the zeros of a complex function in PyLab, you can use the fsolve function from the scipy.optimize module. This function uses a numerical method to approximate the roots of a given function. Alternatively, you can also use the roots function from the numpy module to find the roots of a polynomial function.

4. What is the significance of the zeros of a complex function?

The zeros of a complex function can have multiple interpretations and applications. They can represent the points where the function intersects the x-axis, the solutions to a system of equations, or the critical points of a function. In addition, the location and properties of the zeros can also provide insights into the behavior of the function, such as its symmetry and periodicity.

5. Can a complex function have multiple zeros?

Yes, a complex function can have multiple zeros. In fact, most complex functions have an infinite number of zeros. This is because the complex plane is continuous, meaning that there is no gap between any two complex numbers, and therefore, there is always a value of z for which f(z) = 0. However, some complex functions may have a finite number of distinct zeros, which can be found by solving the equation f(z) = 0.

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