# AEP 4830 HW3 Root Finding and Special Functions

## Plotting of Bessel Functions

The Bessel Functions of the first kind $$J_{\nu}(x)$$ and the second kind $$Y_{\nu}(x)$$ are two functions of interest in this program. The program incorporates two header files ”nr3.h” and ”bessel.h” and outputs the function values between $$x=0$$ to $$x=20$$. The first two Bessel functions $$J_{0}$$, $$J_{1}$$, $$Y_{0}$$ and $$Y_{1}$$ are built in within ”bessel.h” and we can use the recurrence formula to construct higher order Bessel functions.

$$J_{\nu+1}(x)=\frac{2\nu}{x}J_{\nu}(x)-J_{\nu-1}(x)\\$$ $$Y_{\nu+1}(x)=\frac{2\nu}{x}Y_{\nu}(x)-Y_{\nu-1}(x)\\$$

The first three $$J_{\nu}$$ and $$Y_{\nu}$$ are plotted by MATLAB as shown in Fig. 1 and Fig. 2.

The first three Bessel functions of the first kind, $$J_{\nu}(x)$$.

The first three Bessel functions of the second kind, $$Y_{\nu}(x)$$.