Bessel Functions
Let $v$ be a real or integer number and $x$ a positive real number, this calculator evaluates the following six functions.
- The Bessel function of the first kind \[ J_v(x)=\left(\frac{x}{2}\right)^v\sum_{k=0}^\infty\frac{\left(-\frac{x^2}{4}\right)^k}{k!\Gamma(v+k+1)}\]
- The Bessel function of the second kind \[ Y_v(x) = \frac{J_v(x)\cos(v\pi)-J_{-v}(x)}{\sin(v\pi)} \]
- The modified Bessel function of the first kind \[ I_v(x)=\left(\frac{x}{2}\right)^v\sum_{k=0}^\infty\frac{\left(\frac{x^2}{4}\right)^k}{k!\Gamma(v+k+1)}\]
- The modified Bessel function of the second kind \[ K_v(x) = \frac{\pi}{2}\frac{I_{-v}(x)-I_v(x)}{\sin(v\pi)} \]
- (If $v$ is a non-negative integer), the spherical Bessel function of the first kind \[ j_v(x)=\sqrt{\frac{\pi}{2x}}J_{v+\frac{1}{2}}(x)\]
- (If $v$ is a non-negative integer), the spherical Bessel function of the second kind \[ y_v(x)=\sqrt{\frac{\pi}{2x}}Y_{v+\frac{1}{2}}(x)\]