Bessel Functions of the First Kind

This entry provides a formalisation of the Bessel function of the first kind $J_a$ and the modified Bessel function of the first kind $I_a$ as well as several related notions such as Bessel polynomials and the spherical Bessel functions.

Properties that are proven about $I_a$ and $J_a$ include:

• the definition as a hypergeometric series \[\sum_{n\geq 0} (-1)^n (z/2)^{a+2n} / (n! \Gamma(1+a+n))\]
• monotonicity and convexity of $I_a$ on the reals
• the contiguous relations, e.g. $2 a J_a(z) = z (J_{a-1}(z) + J_{a+1}(z))$
• the derivatives, e.g. $J_a' = \frac{1}{2} (J_{a-1} - J_{a+1})$
• closed form expressions for the half-integer case $a = n + \frac{1}{2}$ in terms of trigonometric and hyperbolic functions
• the representation of $J_a$ as contour integral of the form $\int_{c-i\infty}^{c+i\infty}$