 Modified Bessel function

Modified Bessel functions solveThis is the Bessel differential equation after replacing its argument by $iz$ and using the chain rule. The sign change converts the oscillatory positive-axis behavior into exponentially growing and decaying standard solutions, the Modified Bessel function of the first kind $I_{\nu }$ and the Modified Bessel function of the second kind $K_{\nu }$.^[ref]

At order zero, the regular solution normalized by $I_0(0)=1$ has series $I_0(z)={\sum }_{j\ge 0}(z^2/4{)}^j/(j!{)}^2$. Every displayed term is positive for real $z>0$, in contrast to the alternating zero-order $J_0$ series. Boundary conditions requiring decay at positive infinity select $K_0$ instead; regularity at zero and decay at infinity are different choices.