In applied mathematics, a transcendental equation is an equation over the real (or complex) numbers that is not algebraic, that is, if at least one of its sides describes a transcendental function. Examples include: A transcendental equation need not be an equation between elementary functions, although most published examples are. In some cases, a transcendental equation can be solved by transforming it into an equivalent algebraic equation. Some such transformations are sketched below; computer algebra systems may provide more elaborated transformations. In general, however, only approximate solutions can be found.

Transformation into an algebraic equation

Ad hoc methods exist for some classes of transcendental equations in one variable to transform them into algebraic equations which then might be solved.

Exponential equations

If the unknown, say x, occurs only in exponents:

Logarithmic equations

If the unknown x occurs only in arguments of a logarithm function:

Trigonometric equations

If the unknown x occurs only as argument of trigonometric functions:

Hyperbolic equations

If the unknown x occurs only in linear expressions inside arguments of hyperbolic functions,

Approximate solutions

Approximate numerical solutions to transcendental equations can be found using numerical, analytical approximations, or graphical methods. Numerical methods for solving arbitrary equations are called root-finding algorithms. In some cases, the equation can be well approximated using Taylor series near the zero. For example, for k \approx 1, the solutions of are approximately those of, namely x=0 and. For a graphical solution, one method is to set each side of a single-variable transcendental equation equal to a dependent variable and plot the two graphs, using their intersecting points to find solutions (see picture).

Other solutions