Closed-form solutions for voltage-step response of open and shorted distributed RC lines

The current response of a finite distributed RC line to a voltage-step driving function has a closed-form solution in terms of elliptic theta functions. This seems to have gone unrecognized in previous circuit theory literature. Different types of theta functions will appear depending on whether the solution is for open-circuited or short-circuited terminations. Furthermore, the voltage response to the same driving function can be expressed in terms of integrals of theta functions. These integrals constitute a possibly new set of higher transcendental functions, to be denoted here as "diffusion functions." Relationships among and properties of the various solutions and functions are given in mathematical identities and families of curves. Thus expressing the solutions in terms of individual functions whose properties are well-documented, rather than using the mostly opaque series form, allows a more distinct view of the physical behavior of the distributed RC line to emerge.