# Bessel functions of imaginary order

• A
• KariK
In summary, the Wikipedia article on Bessel functions discusses the integral definition of "non-integer order" and its use for imaginary orders. It also mentions the Neumann function and its relation to imaginary orders. The article concludes that for Hankel functions, H^1 and its complex conjugate are linearly independent for imaginary orders. This is shown through the differential equation and its solutions.
KariK
TL;DR Summary
Is it true that the complex conjugate of Hankel function (H) of first kind of order ia is equal to H of second kind of order -ia, so H of first kind and its complex conjugate are linearly independent.
In Wikipedia article on Bessel functions there is an integral definition of “non-integer order” a (“alpha”). For imaginary order ia I get that Jia* = J-ia, where * is complex conjugate and ia and -ia are subscripts. Then in same article there is a definition of Neumann function, again for “non-integer order”. If I again use it for imaginary order, I get that Yia* = Y-ia. So then for Hankel functions I get that H^1ia* = H^2-ia.

Is that valid for imaginary orders and does it then mean that H^1 and its complex conjugate are linearly independent? If the above is not valid, is there some other way to show the independence?

Bessel function Wiki

Yes, they are linearly independent. For integral and half integral ##\alpha## the Hankel are the radial wave functions for incoming and outgoing waves. In the differential equation (first equation in the linked article) ##\alpha## appears as ##\alpha^2##. This implies several things about the equations solutions. First, solutions are analytic functions of ##\alpha## and second, both ##y_\alpha(x)## and ##y_{-\alpha}(x)## obey the very same differential equation. Being a linear second order differential, the are at most two linearly independent solutions, all others expressible as a linear combination.

topsquark
Thank you jedishrfu for moving this thread to a more appropriate place. While I got the Bessel equation from physics, the topic is indeed more math than physics. And thank you Paul Colby for confirming my assumptions.

jedishrfu

## What are Bessel functions of imaginary order?

Bessel functions of imaginary order are a special type of mathematical function that are used to solve differential equations in physics and engineering. They are named after the mathematician Friedrich Bessel and are denoted by the symbol Jν(x), where ν is the order and x is the argument.

## What is the difference between Bessel functions of real and imaginary order?

The main difference between Bessel functions of real and imaginary order is that the order of imaginary Bessel functions can take on complex values, while the order of real Bessel functions is limited to positive or negative integers. Additionally, the behavior of the two types of Bessel functions is different for different values of the argument x.

## What are the applications of Bessel functions of imaginary order?

Bessel functions of imaginary order have many practical applications in physics and engineering. They are used to solve problems involving wave propagation, heat transfer, and diffusion in cylindrical and spherical coordinates. They also have applications in signal processing, quantum mechanics, and electromagnetics.

## How are Bessel functions of imaginary order calculated?

Bessel functions of imaginary order can be calculated using various methods, such as power series, continued fractions, and recurrence relations. They can also be computed using special functions libraries in mathematical software programs like MATLAB or Mathematica.

## What are some properties of Bessel functions of imaginary order?

Bessel functions of imaginary order have many interesting properties, such as orthogonality, recurrence relations, and asymptotic behaviors. They also have connections to other special functions, such as the modified Bessel functions and the hypergeometric functions. These properties make them a powerful tool in solving complex mathematical problems.

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