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Transcendental equation

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

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.[1] 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.[a]

In general, however, only approximate solutions can be found.[2]

Transformation into an algebraic equation[edit]

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[edit]

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

transforms to , which simplifies to , which has the solutions
This will not work if addition occurs "at the base line", as in
  • if all "base constants" can be written as integer or rational powers of some number q, then substituting y=qx may succeed, e.g.
transforms, using y=2x, to which has the solutions , hence is the only real solution.[4]
This will not work if squares or higher power of x occurs in an exponent, or if the "base constants" do not "share" a common q.
  • 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,[citation needed] e.g.:
transforms to which has the solutions hence , where and the denote the real-valued branches of the multivalued function.

Logarithmic equations[edit]

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

  • applying exponentiation to both sides may yield an algebraic equation, e.g.
transforms, using exponentiation to base to which has the solutions If only real numbers are considered, is not a solution, as it leads to a non-real subexpression in the given equation.
This requires the original equation to consist of integer-coefficient linear combinations of logarithms w.r.t. a unique base, and the logarithm arguments to be polynomials in x.[5]
  • if all "logarithm calls" have a unique base and a unique argument expression then substituting may lead to a simpler equation,[6] e.g.
transforms, using to which is algebraic and has the single solution .[b] After that, applying inverse operations to the substitution equation yields

Trigonometric equations[edit]

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,[7] e.g.
transforms to , and, after substitution, to which is algebraic[c] and can be solved. After that, applying obtains the solutions.

Hyperbolic equations[edit]

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,[8] e.g.
unfolds to which transforms to the equation which is algebraic[d] and can be solved. Applying obtains the solutions of the original equation.

Approximate solutions[edit]

Graphical solution of sin(x)=ln(x)

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 , 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 (see picture).

Other solutions[edit]

  • Some transcendental systems of high-order equations can be solved by “separation” of the unknowns, reducing them to algebraic equations.[9][10]
  • The following can also be used when solving transcendental equations/inequalities: If is a solution to the equation and , then this solution must satisfy . For example, we want to solve . The given equation is defined for . Let and . It is easy to show that and so if there is a solution to the equation, it must satisfy . From we get . Indeed, and so is the only real solution to the equation.

See also[edit]


  1. ^ For example, according to the Wolfram Mathematica tutorial page on equation solving, both and can be solved by symbolic expressions, while can only be solved approximatively.
  2. ^ Squaring both sides obtains which has the additional solution ; however, the latter does not solve the unsquared equation.
  3. ^ over an appropriate field, containing and
  4. ^ over an appropriate field, containing


  1. ^ I.N. Bronstein and K.A. Semendjajew and G. Musiol and H. Mühlig (2005). Taschenbuch der Mathematik (in German). Frankfurt/Main: Harri Deutsch. Here: Sect., p.45. The domain of equations is left implicit throughout the book.
  2. ^ Bronstein et al., p.45-46
  3. ^ Bronstein et al., Sect., p.46
  4. ^ Bronstein et al., Sect., p.46
  5. ^ Bronstein et al., Sect., p.46
  6. ^ Bronstein et al., Sect., p.46
  7. ^ Bronstein et al., Sect., p.46-47
  8. ^ Bronstein et al., Sect., p.47
  9. ^ 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
  10. ^ V.A. Varyukhin, Fundamental Theory of Multichannel Analysis (VA PVO SV, Kyiv, 1993) [in Russian]
  • John P. Boyd (2014). Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series, and Oracles. Other Titles in Applied Mathematics. Philadelphia: Society for Industrial and Applied Mathematics (SIAM). doi:10.1137/1.9781611973525. ISBN 978-1-61197-351-8.