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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:
- applying the natural logarithm to both sides may yield an algebraic equation,[3] e.g.
- 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]![](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Sin_x_%3D_ln_x_svg.svg/250px-Sin_x_%3D_ln_x_svg.svg.png)
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]- Mrs. Miniver's problem – Problem on areas of intersecting circles
Notes
[edit]- ^ 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.
- ^ Squaring both sides obtains which has the additional solution ; however, the latter does not solve the unsquared equation.
- ^ over an appropriate field, containing and
- ^ over an appropriate field, containing
References
[edit]- ^ 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.1.6.4.1, p.45. The domain of equations is left implicit throughout the book.
- ^ Bronstein et al., p.45-46
- ^ Bronstein et al., Sect.1.6.4.2.a, p.46
- ^ Bronstein et al., Sect.1.6.4.2.b, p.46
- ^ Bronstein et al., Sect.1.6.4.3.b, p.46
- ^ Bronstein et al., Sect.1.6.4.3.a, p.46
- ^ Bronstein et al., Sect.1.6.4.4, p.46-47
- ^ Bronstein et al., Sect.1.6.4.5, p.47
- ^ 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
- ^ 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.