Transcendental equation
Encyclopedia
A transcendental equation is an equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

 containing a transcendental function
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...

. Examples of such an equation are

Solution methods

Some methods of finding solutions to a transcendental equation use graphical or numerical methods.

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
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...

, using their intersecting points to find solutions.

The numerical solution extends from finding the point at which the intersections occur using some kind of numerical calculation. The solution of transcendental equation obtained by numerical methods are approximate solutions. Approximations can also be made by truncating the Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

 if the variable is considered to be small. Additionally, the fixed point iteration method,the Bisection method
Bisection method
The bisection method in mathematics is a root-finding method which repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow...

, the method of false position, linear interpolation
Linear interpolation
Linear interpolation is a method of curve fitting using linear polynomials. Lerp is an abbreviation for linear interpolation, which can also be used as a verb .-Linear interpolation between two known points:...

, the method of chords, the method of Proportional parts, the Newton-Raphson method or method of tangents could be used to solve the equation.

Often special functions
Special functions
Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications....

 can be used to write the solutions to transcendental equations in closed form
Closed form
-Maths:* Closed-form expression, a finitary expression* Closed differential form, a differential form \alpha with the property that d\alpha = 0-Poetry:* In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern;...

. In particular, the first equation has a solution in terms of the Lambert W Function.
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