Bessel function derivative

In summary: K_0\left(\frac{2\gamma}{\sqrt{p}} \right)In summary, the value of the given derivative is:\frac{4\gamma}{p}e^{-\gamma\sigma/p}K_0\left(\frac{2\gamma}{\sqrt{p}}\right)+\frac{2\gamma\sigma}{p\sqrt{p}}e^{-\gamma\sigma/p}K_1\left(\frac{2\gamma}{\sqrt{p}}\right)which can be obtained by using the product rule and the chain rule for derivatives. I
  • #1
EngWiPy
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Hello,
What is the value of the following derivavtive:
[tex]\frac{d}{d\gamma}\left[ 1-\frac{2\gamma}{\sqrt{p}}e^{-\gamma \sigma/p} K_1\left(\frac{2\gamma}{\sqrt{p}} \right) \right][/tex]​
where [tex]K_1(.)[/tex] is the modified Bessel function of the second kind and order 1?

Some Paper shows that the result is:
[tex]\frac{4\gamma}{p}e^{-\gamma\sigma/p}K_0\left(\frac{2\gamma}{\sqrt{p}}\right)+\frac{2\gamma\sigma}{p\sqrt{p}}e^{-\gamma\sigma/p}K_1\left(\frac{2\gamma}{\sqrt{p}}\right)[/tex]​

But I really don't know how to connect them. The problem is how to handle the Bessel functions in the derivative operation? And what identity must use?

Thanks in advance.
 
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  • #2



Thank you for your question. The value of the derivative you provided can be calculated using the product rule and the chain rule for derivatives. We can rewrite the expression as follows:

\frac{d}{d\gamma}\left[ 1-\frac{2\gamma}{\sqrt{p}}e^{-\gamma \sigma/p} K_1\left(\frac{2\gamma}{\sqrt{p}} \right) \right] = \frac{d}{d\gamma}\left[ 1 \right] - \frac{2}{\sqrt{p}}e^{-\gamma \sigma/p} K_1\left(\frac{2\gamma}{\sqrt{p}} \right) -\frac{2\gamma}{\sqrt{p}}e^{-\gamma \sigma/p} \frac{d}{d\gamma}\left[K_1\left(\frac{2\gamma}{\sqrt{p}} \right)\right]

Using the product rule, we can simplify the last term as:

\frac{d}{d\gamma}\left[K_1\left(\frac{2\gamma}{\sqrt{p}} \right)\right] = \frac{2}{\sqrt{p}}K_0\left(\frac{2\gamma}{\sqrt{p}} \right) - \frac{4\gamma}{p}K_1\left(\frac{2\gamma}{\sqrt{p}} \right)

Substituting this back into the original expression, we get:

\frac{d}{d\gamma}\left[ 1-\frac{2\gamma}{\sqrt{p}}e^{-\gamma \sigma/p} K_1\left(\frac{2\gamma}{\sqrt{p}} \right) \right] = -\frac{2}{\sqrt{p}}e^{-\gamma \sigma/p} K_1\left(\frac{2\gamma}{\sqrt{p}} \right) - \frac{4\gamma}{p}e^{-\gamma \sigma/p}K_0\left(\frac{2\gamma}{\sqrt{p}} \right) + \frac{8\gamma^2}{p^2}e^{-\gamma \sigma/p}K_1\left(\frac{2\gamma}{\sqrt{p}} \right)

 

What is a Bessel function derivative?

A Bessel function derivative is a mathematical function that represents the rate of change of a Bessel function with respect to its argument. Bessel functions are a type of special functions that arise in many areas of physics and engineering, particularly in problems that involve circular or cylindrical symmetry.

What is the physical significance of the Bessel function derivative?

The Bessel function derivative has many physical applications, such as in the study of heat conduction, electromagnetic fields, and quantum mechanics. It is also used in signal processing and image analysis to describe the behavior of circular and cylindrical objects.

How is the Bessel function derivative calculated?

The Bessel function derivative can be calculated using various mathematical techniques, such as the power series method or the integral representation. It can also be expressed in terms of other special functions, such as the Gamma function or the modified Bessel function.

What are the properties of the Bessel function derivative?

The Bessel function derivative has many important properties, such as being an even function, having zeros at certain values of its argument, and exhibiting asymptotic behavior for large and small arguments. It also satisfies a second-order differential equation known as the Bessel equation.

What are some real-world applications of the Bessel function derivative?

The Bessel function derivative has numerous real-world applications, including in the fields of acoustics, optics, and fluid dynamics. It is used to model the behavior of sound waves, light waves, and the flow of fluids in circular or cylindrical domains. It is also used in the design and analysis of various mechanical and electrical systems.

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