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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:
|(for an integer)|
|equivalent to (using the double-angle formula), whose solutions are those of and of , namely and and (for integers)|
Some can be solved because they are compositions of algebraic functions with transcendental functions.
|solve , giving or , then , so or|
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.
|No real solutions, as for all|
|is the only real solution|
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.
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
- 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]