Evaluate Integral of Bessel K Function

In summary, the integral from zero to infinity of xλ * cos(2ax) * [Kv(x)]2 dx, where Kv(x) is the modified Bessel function of the second kind of order v and argument x, has been found to have the form of a hypergeometric function 2F1 by plugging it into Mathematica. The result is given by the ConditionalExpression [(Pi^2*Hypergeometric2F1[5/6, 7/6, 4/3, -a^2])/(4*Gamma[1/3]), Abs[Im[a]] <= 1], with the condition that the absolute value of the imaginary component of a is less than or equal to
  • #1
kschau
1
0
Hey All

Got a tough one and I'm just not seeing the path here. I need to find the close form expression of:

The integral from zero to infinity:

∫xλ * cos(2ax) * [Kv(x)]2 dx

where Kv(x) is the modified Bessel function of the second kind of order v and argument x. If it helps, the value of v=1/3 and the value of λ=2/3

The result will have the form of a hypergeometric function 2F1

I've just been racking my brain for too long with this one. If anyone has some experience with Bessel functions, any help would be appreciated.
 
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  • #2
Have you tried plugging it into Mathematica ?
 
  • #3
In[1]:= Integrate[x^(2/3)*Cos[2 a x]*BesselK[1/3, x]^2, {x, 0, Infinity}]

Out[1]= ConditionalExpression[(Pi^2*Hypergeometric2F1[5/6, 7/6, 4/3, -a^2])/(4*Gamma[1/3]), Abs[Im[a]] <= 1]

so if the absolute value of the imaginary component of a is less than or equal to 1 then you have your hypergeometric as expected.

Verify this independently several different ways before you depend on it.

http://reference.wolfram.com/mathematica/ref/Hypergeometric2F1.html
http://reference.wolfram.com/mathematica/ref/Gamma.html
http://reference.wolfram.com/mathematica/ref/ConditionalExpression.html
 
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What is the Bessel K function?

The Bessel K function is a special function in mathematics that is used to solve differential equations with circular or cylindrical symmetry. It is named after the mathematician Friedrich Bessel and is denoted as Kν(x), where ν is a parameter.

What is the integral of the Bessel K function?

The integral of the Bessel K function can be expressed as ∫Kν(x)dx = -Kν+1(x) + C, where C is a constant of integration. This integral can also be written in terms of the modified Bessel function of the second kind, Kν(x), as ∫Kν(x)dx = xKν(x) + C.

How is the Bessel K function evaluated?

The Bessel K function can be evaluated using various methods, such as the power series method, the continued fraction method, and the asymptotic expansion method. These methods involve manipulating the integral and using mathematical properties of the Bessel K function to simplify the evaluation process.

What are the applications of the Bessel K function?

The Bessel K function has various applications in physics, engineering, and other fields. It is used to solve problems involving heat transfer, quantum mechanics, signal processing, and more. It is also used in the analysis of cylindrical and spherical systems in mathematics.

Are there any special properties of the Bessel K function?

Yes, the Bessel K function has many special properties, such as its logarithmic singularity at x=0 and its asymptotic behavior for large values of x. It also has a relationship with other special functions, such as the modified Bessel function of the first kind and the Hankel function. These properties are important in the evaluation and applications of the Bessel K function.

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