Transcendental equation

A transcendental equation is an equation containing a transcendental function of the variable(s) being solved for. Such equations often do not have closed-form solutions. Examples include:

${\displaystyle {\begin{aligned}x&=e^{-x}\\x&=\cos x\\2^{x}&=x^{2}\end{aligned}}}$

John Herschel, Description of a machine for resolving by inspection certain important forms of transcendental equations, 1832

Solvable transcendental equations

Equations, where the variable to be solved for, appears only once, as an argument to the transcendental function, are easily solvable with inverse functions; similarly, if the equation can be factored or transformed to such a case:

Equation	Solutions
${\displaystyle \ln x=3}$	${\displaystyle x=e^{3}}$
${\displaystyle \sin x=0}$	${\displaystyle x=\pi n}$ (for ${\displaystyle n}$ an integer)
${\displaystyle \cos x=\sin {2x}}$	equivalent to ${\displaystyle \cos x=2\sin x\cos x}$ (using the double-angle formula i.e. sin(2x) = 2cos(x)sin(x)), whose solutions are those of ${\displaystyle \cos x=0}$ and of ${\displaystyle 2\sin x=1}$, namely ${\displaystyle x=\pi n+\pi /2}$ and ${\displaystyle x={2\pi m}+\pi /6}$ and ${\displaystyle x=\pi (2k+1)-\pi /6}$ (for ${\displaystyle m,n,k}$ integers)

Some can be solved because they are compositions of algebraic functions with transcendental functions.

Equation	Solutions
${\displaystyle 3(2^{x})-2=4^{x}}$	solve ${\displaystyle 3q-2=q^{2}}$, giving ${\displaystyle q=1}$ or ${\displaystyle q=2}$, then ${\displaystyle 2^{x}=q}$, so ${\displaystyle x=0}$ or ${\displaystyle x=1}$

But most equations where the variable appears both as an argument to a transcendental function and elsewhere in the equation are not solvable in closed form, or have only trivial solutions.

Equation	Solutions
${\displaystyle e^{x}=x}$	No real solutions, as ${\displaystyle e^{x}>x}$ for all ${\displaystyle x}$
${\displaystyle \sin x=x}$	${\displaystyle x=0}$ is the only real solution

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 ${\displaystyle k\approx 1}$, the solutions of ${\displaystyle \sin x=kx}$ are approximately those of ${\displaystyle (1-k)x-x^{3}/6=0}$, namely ${\displaystyle x=0}$ and ${\displaystyle x=\pm {\sqrt {6}}{\sqrt {1-k}}}$.

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.

In some cases, special functions can be used to write the solutions to transcendental equations in closed form. In particular, ${\displaystyle x=e^{-x}}$ has a solution in terms of the Lambert W function.

Other solutions

The difficulties arising at the solution of the transcendental systems of high-order equations were overcome by Vladimir Varyukhin by means of the “separation” of the unknowns, at which the determination of unknowns is reduced to the solution of algebraic equations[1][2]

References

1. V. A. Varyuhin, S. A. Kas'yanyuk, “On a certain method for solving nonlinear systems of a special type”, Zh. Vychisl. Mat. Mat. Fiz., 6:2 (1966), 347–352; U.S.S.R. Comput. Math. Math. Phys., 6:2 (1966), 214–221
2. V.A. Varyukhin, Fundamental Theory of Multichannel Analysis (VA PVO SV, Kyiv, 1993) [in Russian]