By Mehler's formula, ones of order (*) zero are proportional to the integral from 0 to infinity of the sin of x cosh t dt, while the ones of order plus or minus one-half are equal to the square root of 2 over pi x, times cos or sin
Miller's algorithm is sometimes used to generate a recurrence relation for these entities, and Steed's method simultaneously calculates both them and their derivatives. e to the i z cos θ is equal to the sum from n equals negative to positive infinity of i to the n times e to the i n θ times this function of z by the Jacobi-Anger expansion. By Mehler's formula, ones of order (*) zero are proportional to the integral from 0 to infinity of the sin of x cosh t dt, while the ones of order plus or minus one-half are equal to the square root of 2 over pi x, times cos or sin x. They are the solutions to the equation x squared d squared y d x squared plus x dy dx plus y times quantity x squared minus ν squared equals zero, their namesake's differential equation, but are also used in the solution to the Laplace differential equation in spherical and/or cylindrical coordinates. For 10 points, name these functions denoted by Jν(x) and named after a German astronomer.
Miller's algorithm is sometimes used to generate a recurrence relation for these entities, and Steed's method simultaneously calculates both them and their derivatives.
{Bessel} functions [accept {Bessel functions of the first kind} as all clues point either uniquely to Bessel functions of the first kind or to Bessel functions of any kind] (4) [DW]