Form of Modified Bessel Equation/Functions
The modified Bessel Equation is a type of second-order ODE and a variant of the Bessel Equation (see the last two tutorials) in the form of
\begin{equation}
x^2y^{\prime\prime} + xy’ -(x^2 +\nu^2)y = 0 \tag{1}
\end{equation}
with \( \nu \) as a parameter. The only difference between this and the original Bessel equation is the sign of the \( x^2y \) term. In fact, it can easily be shown that by making the change of variable \( x \to ix \), we can obtain (1) from the original one ((1) of this tutorial):
\begin{align}
(ix)^2\frac{y^{\prime\prime}}{i^2} + (ix)\frac{y’}{i} + ((ix)^2 -\nu^2)y &= 0 \tag{2}
\end{align}
This enables us to immediately write down the solutions to (1) like the \( J_{\nu}(x) \) counterpart ((8) of the aforementioned tutorial):
\begin{align}
y_{\pm \nu} &= \sum_{m=0}^{\infty} \frac{(-1)^m}{m!\Gamma(m+1\pm \nu)} (\frac{ix}{2})^{\pm\nu+2m} \\
&= \sum_{m=0}^{\infty} \frac{(-1)^m(-1)^m}{m!\Gamma(m+1\pm \nu)} (\frac{x}{2})^{\pm\nu+2m} \\
&= \sum_{m=0}^{\infty} \frac{1}{m!\Gamma(m+1\pm \nu)} (\frac{x}{2})^{\pm\nu+2m} \tag{3} \\
\end{align}
and this will be the modified Bessel functions of the first kind of order \( \pm \nu \), denoted as \( I_{\nu}(x) \) and \( I_{-\nu}(x)\). The modified Bessel functions of the second kind \( K_{\nu}(x) \) are also similarly defined like \( Y_{\nu}(x) \). Basically, the modified Bessel functions have similar properties to the original Bessel functions mentioned in the last tutorial, and so we will simply list them below.
Orthogonality and Normalization Factor
Unfortunately, the modified Bessel functions don’t have an orthogonality property and a normalization factor analogous to the ones for the original Bessel functions. In particular, the modified Bessel functions have no zeros for \( x > 0 \).
Recurrence Relations
Similar recurrence relations for modified Bessel Functions (of the first kind) hold:
\begin{align}
\begin{aligned}
\frac{d}{dx}(x^\nu I_\nu(x)) &= x^\nu I_{\nu-1}(x) \\
\frac{d}{dx}(x^{-\nu} I_\nu(x)) &= x^{-\nu} I_{\nu+1}(x)
\end{aligned} \tag{4}
\end{align}
\begin{align}
x I_\nu(x)’ + \nu I_\nu(x) &= x I_{\nu-1}(x) \tag{5} \\
x I_\nu(x)’ -\nu I_\nu(x) &= x I_{\nu+1}(x) \tag{6} \\
2 I_\nu(x)’ &= I_{\nu-1}(x) + I_{\nu+1}(x) \tag{7} \\
2 \nu I_\nu(x) &= x I_{\nu-1}(x) -x I_{\nu+1}(x) \tag{8} \\
\end{align}
Generating Function
The generating function of modified Bessel functions of the first kind with integer order \( n \) closely parallels the old one:
\begin{align}
\exp(\frac{x}{2} (t+\frac{1}{t})) = \sum_{n=-\infty}^{\infty} I_n(x) t^n \tag{9}
\end{align}
Integral Representation
They also have slightly different integral representations, e.g.
\begin{align}
I_n(x) = \frac{1}{\pi} \int_0^\pi e^{x \cos \theta} \cos(n\theta) d\theta \tag{10}
\end{align}
Relation to Confluent Hypergeometric Function
The modified Bessel functions of the first kind can be expressed in terms of the confluent hypergeometric function (see this tutorial):
\begin{align}
I_{\nu}(x) = \frac{e^{-x}}{\Gamma(\nu+1)}M(\nu+\frac{1}{2}, 2\nu+1; 2x) (\frac{x}{2})^{\nu} \tag{11}
\end{align}
Exercise
Show that
\begin{align}
I_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}} \sinh x \tag{12}
\end{align}
Answer
By (3):
\begin{align}
I_{\frac{1}{2}}(x) &= \sum_{m=0}^{\infty} \frac{1}{m!\Gamma(m+1+\frac{1}{2})} (\frac{x}{2})^{\frac{1}{2}+2m} \\
&= \sqrt{\frac{2}{x}} \sum_{m=0}^{\infty} \frac{1}{m!\Gamma(m+1+\frac{1}{2})} (\frac{x}{2})^{2m+1} \\
&= \sqrt{\frac{2}{x}} \sum_{m=0}^{\infty} \frac{1}{(2^mm!)(2^{m+1}\Gamma(m+1+\frac{1}{2}))} x^{2m+1} \\
&= \sqrt{\frac{2}{x}} \sum_{m=0}^{\infty} \frac{1}{(2)(4)\cdots(2m)\Gamma(\frac{1}{2})(1)(3)(5)\cdots(2m+1)} x^{2m+1} \\
&= \sqrt{\frac{2}{\pi x}} \sum_{m=0}^{\infty} \frac{1}{(2m+1)!} x^{2m+1} = \sqrt{\frac{2}{\pi x}} \sinh x
\end{align}
where we have recalled the Taylor series of \( \sinh x \) and \( \Gamma(1/2) = \sqrt{\pi}\).
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