For 10 points, name these mutually-orthogonal functions that solve Laplace's equation in cylindrical coordinates.
The generating function for this function is the exponential of x over 2, times t minus 1 over t. Kummer's confluent hypergeometric function reduces to these functions in the case that b equals 2a. Â The amplitudes of the sidebands of a frequency-modulated signal are given by these functions. This function of order n can be defined as the sum of this function of order n minus 1 and n plus 1, times two x divided by n. The second kind of this function has a singularity at the origin, and is named for (*) Neumann. This function of order at least one-half has at least one zero of the sine function between each of its zeros, and this function of exactly order one-half is a damped sine function. For 10 points, name these mutually-orthogonal functions that solve Laplace's equation in cylindrical coordinates.
The generating function for this function is the exponential of x over 2, times t minus 1 over t. Kummer's confluent hypergeometric function reduces to these functions in the case that b equals 2a.
Bessel functions (of the first kind) [accept cylindrical harmonics; prompt on "cylinder functions"]