Transcendental equation

In applied mathematics, a transcendental equation is an equation over the real 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 algebraic equation|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:

applying the natural logarithm to both sides may yield an algebraic equation, e.g.
if all "base constants" can be written as integer or rational powers of some number q, then substituting y=''qx'' may succeed, e.g.
sometimes, substituting y=''xex'' may obtain an algebraic equation; after the solutions for y are known, those for x can be obtained by applying the Lambert W function, e.g.:

Logarithmic equations

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

applying exponentiation to both sides may yield an algebraic equation, e.g.
if all "logarithm calls" have a unique base and a unique argument expression then substituting may lead to a simpler equation, e.g.

Trigonometric equations

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

applying Pythagorean identities and trigonometric sum and multiple formulas, arguments of the forms with integer might all be transformed to arguments of the form, say,. After that, substituting yields an algebraic equation, e.g.

Hyperbolic equations

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

unfolding them by their defining exponential expressions and substituting yields an algebraic equation, e.g.

Approximate solutions

Approximate numerical solutions to transcendental equations can be found using numerical, analytical approximations, or graphical methods.
These equations can be solved by direct iteration by reordering the equation into the form and
making an initial guess, computing which becomes and substituting it back into, etc. Convergence may be very slow. Some reorderings may diverge, so some other reordering that converges must be found. must be continuous and "sufficiently smooth" or the method may fail.
Numerical methods for solving arbitrary equations are called root-finding algorithms. By rearranging the equation into the form, if is continuous and differentiable, Newton's method involving taking the derivative of, is a common iterative method of approximating a root; an initial guess must be "sufficiently close" to the root of interest to converge to it.
In some cases, the equation can be well approximated using Taylor series near the zero. For example, for, the solutions of are approximately those of, namely 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.