Confluent Hypergeometric Functions
The confluent hypergeometric differential equation has a regular singular point at and an essential singularity at . Solutions analytic at are confluent hypergeometric functions of the first kind (or Kummer functions): , where are Pochhammer symbols defined by , , . For , the function becomes singular, unless is an equal or smaller negative integer (), and it is convenient to define the regularized confluent hypergeometric , which is an entire function for all values of , and . The second, linearly independent solutions of the differential equation are confluent hypergeometric functions of the second kind (or Tricomi functions), defined by , where the generalized hypergeometric function represents a formal asymptotic series. If the hypergeometric function with argument is complex, both the real and imaginary parts are plotted (black and red curves). For certain choices of the parameters and , the hypergeometric functions are related to various transcendental and special functions. Several illustrations are given in the snapshots.
Snapshot 1: when , reduces to an exponential function; for example, , for which the real and imaginary parts are plotted
Snapshot 2: relation to Bessel functions:
Snapshot 3: Laguerre polynomials:
Snapshot 4: error function:
Snapshot 5: incomplete gamma function:
Snapshot 6: modified Bessel function: