Bessel Functions

The Bessel functions are rather complicated oscillatory functions with many interesting properties. The functions Iv(z) and Kv(z) are solutions of the first and second kind, respectively, to the modified Bessel equation

z2$\displaystyle {\frac{{d^{2}w}}{{dz^{2}}}}$ + z$\displaystyle {\frac{{dw}}{{dz}}}$ - $\displaystyle \left(\vphantom{ z^{2}+v^{2}}\right.$z2 + v2$\displaystyle \left.\vphantom{ z^{2}+v^{2}}\right)$w = 0

and the functions Jv(z) and Yv(z) are solutions of the first and second kind, respectively, to the Bessel equation

z2$\displaystyle {\frac{{d^{2}w}}{{dz^{2}}}}$ + z$\displaystyle {\frac{{dw}}{{dz}}}$ + $\displaystyle \left(\vphantom{ z^{2}-v^{2}}\right.$z2 - v2$\displaystyle \left.\vphantom{ z^{2}-v^{2}}\right)$w = 0

Following are plots of these functions.

itbpFU3.0441in2.0384in0inIv(z)besseli.wmf itbpFU3.0441in2.0384in0inKv(z)besselk.wmf

itbpFU3.0441in2.0384in0inJv(z)besselj.wmf itbpFU3.0441in2.0384in0inYv(z)bessely.wmf