Bessel Functions
Calculates the values of the first-kind Bessel function Jv(x), the second-kind Bessel function Yv(x), and their derivatives J'v(x), Y'v(x).
The variable x accepts complex numbers.
| \(\normalsize Bessel\ functions\ of\\ \ the\ 1st\ kind\ J_\nu(x)\ and\ 2nd\ kind\ Y_\nu(x)\\ (1)\ x^2y''+xy'+(x^2-\nu^2)y=0\\ \hspace{25px} y=c_1J_\nu(x)+c_2Y_\nu(x)\\ (2)\ J_\nu(x)={\large\displaystyle \sum_{\small k=0}^{\small\infty}\frac{(-1)^k}{k!\Gamma(k+\nu+1)}(\frac{x}{2})^{\nu+2k}}\\ \hspace{30px} Y_\nu(x)={\large\frac{J_\nu(x) \cos(\nu\pi)-J_{-\nu}(x)}{\sin(\nu\pi)}}\\ (3)\ J'_\nu(x)=J_{\nu-1}(x)-{\large\frac{\nu}{x}}J_\nu(x)=-J_{\nu+1}(x)+{\large\frac{\nu}{x}}J_\nu(x)\\ \hspace{25px}Y'_\nu(x)=Y_{\nu-1}(x)-{\large\frac{\nu}{x}}Y_\nu(x)=-Y_{\nu+1}(x)+{\large\frac{\nu}{x}}Y_\nu(x)\\ (4)\ {\large e^{\frac{x}{2}(t-{\large\frac{1}{t}})}}={\large\displaystyle \sum_{\small n=-\infty}^{\small\infty}}J_n(x)t^n,\hspace{20px} n=integer\\\) |
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