Generalized Confluent Hypergeometric and Hypergeometric Transmission Lines

A proliferation of exact closed-form solutions of the telegrapher's equation
V_{xx} - Z_{x}Z^{-1}V_{x} - kZ Y V = 0
for the voltage
V(x)
in an
RC
or lossless transmission line, with distributed series impedance
Z(x)
and shunt admittance
Y(x)
, respectively, have emerged in recent years. Generalizations of known solutions have been constructed, sometimes using ad hoc methods. A systematic method is described for deriving exact solutions in terms of standard transcendental functions, which yields far more general profiles for
Z(x), Y(x)
, or
Z(x)/ Y(x)
than previously given. Examples of the procedure are given based upon Bessel's, Whittaker's, and the hypergeometric equation, and previously derived profiles emerge as special cases of the analysis.