He began publishing papers on a new topic, algebraic functions, which would prove to be the most fruitful research field of his career.
In the attack, an algebraic function is used to represent an S-box.
Thus an algebraic function is most naturally considered as a multiple valued function.
Thus, problems to do with the domain of an algebraic function can safely be minimized.
On a more significant theoretical level, using complex numbers allows one to use the powerful techniques of complex analysis to discuss algebraic functions.
Of course, this reduction can take place only if the integral is an algebraic function not very complicated with respect to its variables.
A "signomial" is an algebraic function of one or more independent variables.
A simple example of an algebraic function is given by the unit circle equation:
The natural logarithm is a transcendental function, an entity beyond the class of algebraic functions.
He also taught courses on algebraic functions and abelian integrals.