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Transcendental equation - Wikipedia

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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:

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

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

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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

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

{\displaystyle 4^{x}=3^{x^{2}-1}\cdot 2^{5x}} transforms to {\displaystyle x\ln 4=(x^{2}-1)\ln 3+5x\ln 2}, which simplifies to {\displaystyle x^{2}\ln 3+x(5\ln 2-\ln 4)-\ln 3=0}, which has the solutions {\displaystyle x={\frac {-3\ln 2\pm {\sqrt {9(\ln 2)^{2}-4(\ln 3)^{2}}}}{2\ln 3}}.}
This will not work if addition occurs "at the base line", as in {\displaystyle 4^{x}=3^{x^{2}-1}+2^{5x}.}
  • if all "base constants" can be written as integer or rational powers of some number q, then substituting y=qx may succeed, e.g.
{\displaystyle 2^{x-1}+4^{x-2}-8^{x-2}=0} transforms, using y=2x, to {\displaystyle {\frac {1}{2}}y+{\frac {1}{16}}y^{2}-{\frac {1}{64}}y^{3}=0} which has the solutions {\displaystyle y\in \{0,-4,8\}}, hence {\displaystyle x=\log _{2}8=3} 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.:
{\displaystyle x^{2}e^{2x}+2=3xe^{x}} transforms to {\displaystyle y^{2}+2=3y,} which has the solutions {\displaystyle y\in \{1,2\},} hence {\displaystyle x\in \{W_{0}(1),W_{0}(2),W_{-1}(1),W_{-1}(2)\}}, where {\displaystyle W_{0}} and {\displaystyle W_{-1}} denote the real-valued branches of the multivalued {\displaystyle W} function.

Logarithmic equations

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

  • applying exponentiation to both sides may yield an algebraic equation, e.g.
{\displaystyle 2\log _{5}(3x-1)-\log _{5}(12x+1)=0} transforms, using exponentiation to base {\displaystyle 5.} to {\displaystyle {\frac {(3x-1)^{2}}{12x+1}}=1,} which has the solutions {\displaystyle x\in \{0,2\}.} If only real numbers are considered, {\displaystyle x=0} is not a solution, as it leads to a non-real subexpression {\displaystyle \log _{5}(-1)} 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]
{\displaystyle 5\ln(\sin x^{2})+6=7{\sqrt {\ln(\sin x^{2})+8}}} transforms, using {\displaystyle y=\ln(\sin x^{2}),} to {\displaystyle 5y+6=7{\sqrt {y+8}},} which is algebraic and has the single solution {\displaystyle y={\frac {89}{25}}}.[b] After that, applying inverse operations to the substitution equation yields {\displaystyle x={\sqrt {\arcsin \exp y}}={\sqrt {\arcsin \exp {\frac {89}{25}}}}.}

Trigonometric equations

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

{\displaystyle \sin(x+a)=(\cos ^{2}x)-1} transforms to {\displaystyle (\sin x)(\cos a)+{\sqrt {1-\sin ^{2}x}}(\sin a)=1-(\sin ^{2}x)-1}, and, after substitution, to {\displaystyle y(\cos a)+{\sqrt {1-y^{2}}}(\sin a)=-y^{2}} which is algebraic[c] and can be solved. After that, applying {\displaystyle x=2k\pi +\arcsin y} obtains the solutions.

Hyperbolic equations

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

  • unfolding them by their defining exponential expressions and substituting {\displaystyle y=\exp(x)} yields an algebraic equation,[8] e.g.
{\displaystyle 3\cosh x=4+\sinh(2x-6)} unfolds to {\displaystyle {\frac {3}{2}}(e^{x}+{\frac {1}{e^{x}}})=4+{\frac {1}{2}}\left({\frac {(e^{x})^{2}}{e^{6}}}-{\frac {e^{6}}{(e^{x})^{2}}}\right),} which transforms to the equation {\displaystyle {\frac {3}{2}}(y+{\frac {1}{y}})=4+{\frac {1}{2}}\left({\frac {y^{2}}{e^{6}}}-{\frac {e^{6}}{y^{2}}}\right),} which is algebraic[d] and can be solved. Applying {\displaystyle x=\ln y} obtains the solutions of the original equation.

Approximate solutions

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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 {\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 (see picture).

  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.1.6.4.1, p.45. The domain of equations is left implicit throughout the book.
  2. ^ Bronstein et al., p.45-46
  3. ^ Bronstein et al., Sect.1.6.4.2.a, p.46
  4. ^ Bronstein et al., Sect.1.6.4.2.b, p.46
  5. ^ Bronstein et al., Sect.1.6.4.3.b, p.46
  6. ^ Bronstein et al., Sect.1.6.4.3.a, p.46
  7. ^ Bronstein et al., Sect.1.6.4.4, p.46-47
  8. ^ Bronstein et al., Sect.1.6.4.5, 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.