A transcendental equation is an equation containing transcendental functions. The transcendental function is any function which is the solution of the equation

where P_{i}(x) are polynomials of x.

Elementary transcendental functions are the exponential, logarithmic, trigonometric, reverse trigonometric, and hyperbolic functions. If transcendental functions are considered as functions of a complex variable, then their characteristic feature is the presence of at least one singularity in addition to poles and branch points of finite order. For instance, the functions e^{z}, cos z, sin z have a significant singular point z = ∞, and also branch points of infinite order z = 0 and z = ∞.

An important class of transcendental functions are cylindrical and spherical functions often met in problems of heat transfer, including the gamma and beta functions, Euler's functions, hyper-geometric and degenerate hypergeometric functions. Transcendental equations occur, for instance, on finding the eigenvalues, when solving problems of heat transfer by the method of separation of variables. The transcendental equations are usually solved with the help of numerical methods such as the Newton method, the method of false position, etc.

Heat & Mass Transfer, and Fluids Engineering