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applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical ...
, a transcendental equation is an
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
over the
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
(or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
) numbers that is not algebraic, that is, if at least one of its sides describes a
transcendental function In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed alg ...
. Examples include: :\begin x &= e^ \\ x &= \cos x \\ 2^x &= x^2 \end A transcendental equation need not be an equation between
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exp ...
s, 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 In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressio ...
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 The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
to both sides may yield an algebraic equation, e.g. : 4^x = 3^ \cdot 2^ transforms to x \ln 4 = (x^2-1) \ln 3 + 5x \ln 2, which simplifies to x^2 \ln 3 + x(5 \ln 2 - \ln 4) -\ln 3 = 0, which has the solutions x = \frac . : This will not work if addition occurs "at the base line", as in 4^x = 3^ + 2^ . * if all "base constants" can be written as integer or rational powers of some number ''q'', then substituting ''y''=''q''''x'' may succeed, e.g. : 2^ + 4^ - 8^ = 0 transforms, using ''y''=2''x'', to \frac y + \frac y^2 - \frac y^3 = 0 which has the solutions y \in \, hence x= \log_2 8 = 3 is the only real solution. : 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''=''x''e''x'' may obtain an algebraic equation; after the solutions for ''y'' are known, those for ''x'' can be obtained by applying the
Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential functio ...
, e.g.: : x^2e^ + 2 = 3x e^x transforms to y^2 + 2 = 3y, which has the solutions y \in \, hence x \in \, where W_0 and W_ the denote the real-valued branches of the multivalued W function.


Logarithmic equations

If the unknown ''x'' occurs only in arguments of a
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
function: * applying
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to r ...
to both sides may yield an algebraic equation, e.g. :2 \log_5 (3x-1) - \log_5 (12x+1) = 0 transforms, using exponentiation to base 5. to \frac = 1, which has the solutions x \in \ . If only real numbers are considered, x = 0 is not a solution, as it leads to a non-real subexpression \log_5(-1) in the given equation. :This requires the original equation to consist of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
-coefficient linear combinations of logarithms w.r.t. a unique base, and the logarithm arguments to be polynomials in ''x''. * if all "logarithm calls" have a unique base b and a unique argument expression f(x), then substituting y = \log_b (f(x)) may lead to a simpler equation, e.g. : 5 \ln(\sin x^2) + 6 = 7 \sqrt transforms, using y = \ln(\sin x^2) , to 5 y + 6 = 7 \sqrt, which is algebraic and can be solved. After that, applying inverse operations to the substitution equation yields \sqrt = x.


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 \sin(nx+a), \cos(mx+b), \tan(lx+c), ... with integer n,m,l,... might all be transformed to arguments of the form, say, \sin x. After that, substituting y = \sin(x) yields an algebraic equation, e.g. : \sin(x+a) = (\cos^2 x) - 1 transforms to (\sin x)(\cos a) + \sqrt(\sin a) = 1 - (\sin^2 x) - 1, and, after substitution, to y (\cos a) + \sqrt(\sin a) = - y^2 which is algebraic and can be solved. After that, applying x = 2k\pi + \arcsin y obtains the solutions.


Hyperbolic equations

If the unknown ''x'' occurs only in linear expressions inside arguments of
hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
s, * unfolding them by their defining exponential expressions and substituting y = exp(x) yields an algebraic equation, e.g. : 3 \cosh x = 4 + \sinh (2x-6) unfolds to \frac (e^x + \frac) = 4 + \frac \left( \frac - \frac \right) , which transforms to the equation \frac (y + \frac) = 4 + \frac \left( \frac - \frac \right) , which is algebraic and can be solved. Applying x = \ln y obtains the solutions of the original equation.


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 mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex num ...
. In some cases, the equation can be well approximated using Taylor series near the zero. For example, for k \approx 1, the solutions of \sin x = k x are approximately those of (1-k) x - x^3/6=0, namely x=0 and x = \plusmn \sqrt \sqrt. For a graphical solution, one method is to set each side of a single variable transcendental equation equal to a
dependent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
and plot the two
graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...
, using their intersecting points to find solutions (see picture).


Other solutions

* Some transcendental systems of high-order equations can be solved by “separation” of the unknowns, reducing them to algebraic equations.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), 214–221V.A. Varyukhin, Fundamental Theory of Multichannel Analysis (VA PVO SV, Kyiv, 1993)
n Russian N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...
/ref> * The following can also be used when solving transcendental equations/inequalities: If x_0 is a solution to the equation f(x)=g(x) and f(x)\leq c\leq g(x), then this solution must satisfy f(x_0)=g(x_0)=c. For example, we want to solve \log_\left(3+2x-x^\right)=\tan^\left(\frac\right)+\cot^\left(\frac\right). The given equation is defined for -1. Let f(x)=\log_\left(3+2x-x^\right) and g(x)=\tan^\left(\frac\right)+\cot^\left(\frac\right). It is easy to show that f(x)\leq 2 and g(x)\geq 2 so if there is a solution to the equation, it must satisfy f(x)=g(x)=2. From f(x)=2 we get x=1\in(-1,3). Indeed, f(1)=g(1)=2 and so x=1 is the only real solution to the equation.


See also

*


References

* {{cite book , doi=10.1137/1.9781611973525 , isbn=978-1-61197-351-8 , author=John P. Boyd , editor= , title=Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series, and Oracles , location=Philadelphia , publisher=Society for Industrial and Applied Mathematics (SIAM) , series=Other Titles in Applied Mathematics , volume= , edition= , year=2014 Equations