Integral representation of modified Bessel function of the second kind

In summary, the integral representation of modified Bessel function of the second kind, denoted as K<sub>v</sub>(x), is given by: K<sub>v</sub>(x) = (π/2x)<sup>1/2</sup> ∫<sub>0</sub><sup>∞</sup> e<sup>-x cosh t</sup> cos(vt) dt, where Re(x) > 0 and Re(v) is any complex number. It has various applications in mathematical physics, engineering, and statistics. The main difference from the regular Bessel function is its imaginary argument and broader domain of definition. The integral representation can only be used for specific values of x
  • #1
ulriksvensson
21
0
Hi all. I need an integral representation of [tex]z^{-\nu}K_{\nu}[/tex] of a particular form. For [tex]K_{1/2}[/tex] it looks like this:

[tex]z^{-\frac{1}{4}}K_{1/2}(\sqrt{z}) \propto \int_{0}^{\infty}dt\exp^{-zt-1/t}t^{-1/2}[/tex]

How do I generalize this for arbitrary [tex]\nu[/tex]? A hint is enough, maybe there's a generating function one can use?

//Ulrik
 
Mathematics news on Phys.org
  • #2
Hi
The general form is attached.
 

Attachments

  • bessel integral.pdf
    194.4 KB · Views: 571

FAQ: Integral representation of modified Bessel function of the second kind

What is the integral representation of modified Bessel function of the second kind?

The integral representation of modified Bessel function of the second kind, denoted as Kv(x), is given by:
Kv(x) = (π/2x)1/2 ∫0∞ e-x cosh t cos(vt) dt, where Re(x) > 0 and Re(v) is any complex number.

What is the significance of the modified Bessel function of the second kind?

The modified Bessel function of the second kind has various applications in mathematical physics, engineering, and statistics. It is commonly used in solving differential equations, calculating probability distributions, and describing the behavior of physical systems.

How does the modified Bessel function of the second kind differ from the regular Bessel function?

The main difference between the modified Bessel function of the second kind and the regular Bessel function is that the former has an imaginary argument, while the latter has a real argument. Additionally, the modified Bessel function of the second kind is defined for all complex numbers, while the regular Bessel function is only defined for non-negative integers.

Can the integral representation be used to evaluate Kv(x) for any values of x and v?

No, the integral representation of modified Bessel function of the second kind can only be used to evaluate Kv(x) for specific values of x and v. For example, if x is a negative real number, the integral diverges, and if v is a complex number with a non-zero imaginary part, the integral may not converge.

Are there any other representations of the modified Bessel function of the second kind?

Yes, there are other representations of the modified Bessel function of the second kind, including series representations, continued fraction representations, and asymptotic representations. These representations may be more convenient to use in certain situations, such as for small or large values of x or v.

Similar threads

I Infinite product representation of Bessel's function of the 2nd kind
Replies
2
Views
2K
I Integral representation of incomplete gamma function
Replies
3
Views
2K
Integrals of the Bessel functions of the first kind
Replies
1
Views
1K
Modified Bessel function with imaginary index is purely real?
Replies
1
Views
2K
Sum formula for the modified Bessel function
Replies
1
Views
2K
I Modified Bessel Equation
Replies
3
Views
2K
MHB Exploring the Bessel Function Expansion
Replies
3
Views
3K
Normalization of Bessel functions of the first kind
Replies
1
Views
8K
I Calculating the inverse of a function involving the error function
Replies
3
Views
1K
A Integral of 2 Bessel functions of different orders
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top