📂Functions

Modified Bessel Equation and Modified Bessel Function

Buildup

The differential equation below is referred to as the modified Bessel equation.

$$ x^2 y^{\prime \prime} + xy^{\prime}-(x^2-\nu^2)y=0 $$

It is a form of the Bessel equation where the sign of the term $y$ has been changed to $+ \rightarrow -$. The solution to this differential equation is given by the formula for differential equations that have Bessel equation solutions, as follows.

$$ y=Z_{\nu}(ix)=AJ_{\nu}(ix)+BN_{\nu}(ix) $$

The two commonly used forms of the solution are referred to as modified Bessel functions. In particular, $I_{\nu}$ is called the modified Bessel function of the first kind, and $K_{\nu}$ is called the modified Bessel function of the second kind.

Definition

The modified Bessel function of the first kind $I_{\nu}$ and the modified Bessel function of the second kind $K_{\nu}$ are defined as follows.

$$ \begin{align*} I_{\nu}(x)&=i^{-\nu}J_{\nu}(ix) \\ \\ K_{\nu}(x) &= \frac{\pi}{2}i^{\nu+1}\left[ J_{\nu}(ix)+iN_{\nu}(ix) \right] \\ &= \frac{\pi}{2}i^{\nu+1}H_{p}^{(1)}(ix) \\ &=\frac{\pi}{2}\frac{I_{-\nu}(x)-I_{\nu}(x)}{\sin (\nu\pi )} \end{align*} $$

Here, $J_{\nu}$ $H_{\nu}^{(1)}(x)$ are Hankel functions.

Explanation

The reason for multiplication by $i$ upfront is to ensure that for real $x$, the values of $I_{\nu}(x)$ and $K_{\nu}(x)$ are real. This situation is similar to that in which the solutions of $y^{\prime \prime}+y=0$ are $\cos x$ and $\sin x$, and the solutions of $y^{\prime \prime}-y=0$ are $\cosh x=\cos (ix)$ and $\sinh (x)=\sin (ix)$. Due to these characteristics of the equations, $I_{\nu}$ and $K_{\nu}$ are also referred to as hyperbolic Bessel functions.

Integral Form

An integral form was made known by Olver et al. in 2010¹.

$$ I_{\nu} (z) = {{ \left( {{ z } \over { 2 }} \right)^{\nu} } \over { \sqrt{\pi} \Gamma \left( \nu + {{ 1 } \over { 2 }} \right) }} \int_{-1}^{1} e^{zt} \left( 1 - t^{2} \right)^{\nu - {{ 1 } \over { 2 }}} dt $$

Such modified Bessel functions are crucial not only in mathematical physics but also in areas like directional statistics, and they appear in the Matérn function, which is one of the plausible choices for the semivariogram in spatial statistical analysis.