# Spherical Bessel function

## Calculates the spherical Bessel functions of the first kind jv(x) and the second kind yv(x), and their derivatives j'v(x) and y'v(x).

order v
 real number
x
 complex number

jv(x)
yv(x)
j'v(x)
y'v(x)

 $\normal Spherical\ Bessel\ function\ of\\\ the\ 1st\ kind\ j_\nu(x)\ and\ 2nd\ kind\ y_\nu(x)\\[10](1)\ x^2w''+2xw'+(x^2-\nu(\nu+1))w=0\\\hspace{25} w=c_1j_\nu(x)+c_2y_\nu(x)\\[10](2)\ j_\nu(x)={\large\sqrt{\frac{\pi}{2x}}}J_{\nu+\frac{1}{2}}(x)\\\hspace{25}y_\nu(x)={\large\sqrt{\frac{\pi}{2x}}}Y_{\nu+\frac{1}{2}}(x)\\[10](3)\ j'_\nu(x)=j_{\nu-1}(x)-{\large\frac{\nu+1}{x}}j_\nu(x)=-j_{\nu+1}(x)+{\large\frac{\nu}{x}}j_\nu(x)\\\hspace{25}y'_\nu(x)=y_{\nu-1}(x)-{\large\frac{\nu+1}{x}}y_\nu(x)=-y_{\nu+1}(x)+{\large\frac{\nu}{x}}y_\nu(x)\\$

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