Transcendental Function

Dhristi JEE 2022-24

What are Transcendental Functions?

In mathematics, when a function is not expressible in terms of a finite combination of algebraic operation of addition, subtraction, division, or multiplication raising to a power and extracting a root, then they are said to be transcendental functions. Some of the examples of transcendental functions can be log x, sin x, cos x, etc. These functions that are non-algebraic in nature can only be expressed in terms of infinite series. 


In Mathematics, transcendental functions are the analytical functions that are not algebraic, and hence do not satisfy the polynomial equation. In other words, transcendental functions cannot be expressed in terms of finite sequence of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and extracting the roots. The functions such as logarithmic, trigonometric functions, and exponential functions are a few examples of transcendental functions.  


The transcendental functions can be expressed in algebra only in the terms of an infinite sequence. Hence, the term transcendental means non-algebraic.


Define Transcendental Functions

The transcendental function can be defined as a function that is not algebraic and cannot be expressed in terms of a finite sequence of algebraic operations such as sin x. 


The most familiar transcendental functions examples are the exponential functions, logarithmic functions, trigonometric functions, hyperbolic functions, and inverse of all these functions. The less familiar transcendental functions examples are Gamma, Elliptic, and Zeta functions.


What is a Transcendental Equation?

A polynomial equation is an equation in the form of 

\[x^{4} - 4x^{2} - 3 = 0, 4x^{2} - 3x + 9  = 0, and 2x^{3}- 5x^{2} - 7x + 3\] are some of the algebraic equations.

An equation containing polynomials, logarithmic functions, trigonometric functions, and exponential functions is known as transcendental equation.

\[tan x - e^{x} = 0, sin x - xe^{2x} = 0 , and xe^{x} = cos x\] are some of the transcendental equations examples.


Define Transcendental Equations

A transcendental equation is an equation into which transcendental functions (such as exponential, logarithmic, trigonometric, or inverse trigonometric) of one of the variables (s) have been solved for. Transcendental equations do not have closed-form solutions. 


Transcendental equations examples includes: \[x =e^{-x}, x = cos x, 2^{x} = x^{2}\].


Transcendental Functions Examples With Solutions

1. Find dy/dx for the function y = In(tan x + sec x)

Solution: 

dy/dx = x² (1/4x. 4) + In (4x). 2x

= x + 2x In ( 4x)

= x( 1 + 2 In (4x))


2. Calculate \[\lim_{x\rightarrow 0}\frac{secx-1}{sinx}\]

Solution: 

As both numerator and denominator approaches to 0. Hence, applying L’s hospital rule, we get:

\[\lim_{^{2}x\rightarrow 0}\frac{secx-1}{sinx}\]

\[=\lim_{x\rightarrow 0}\frac{secx-tanx}{cosx}\]

\[=\lim_{x\rightarrow 0}\frac{1.0}{1}\]

= 1

FAQs on Transcendental Function

1. What are the different methods to find algebraic and transcendental equations?

The different methods to find algebraic and transcendental equations are mentioned below:

  • Bisection Method

  • Method of False Position

  • Newton Raphson Method

  • The Iteration Method

2. Which functions are not considered as transcendental functions?

A function that is not transcendental is known as an algebraic function. Rational functions and square root functions are examples of transcendental functions. The indefinite integral of many transcendental functions is considered to be transcendental. For example, logarithmic functions are derived from reciprocal functions with a view to finding the area of the hyperbolic sector. 

3. What is the difference between algebraic equations and transcendental functions?

A function y = f (x) is said to be an algebraic equation if it is a root of a polynomial in y whose coefficients corresponds to polynomial in x or it can also be said that if a function f is called an algebraic function only if it involves the algebraic operation such as addition, multiplication, division, and subtraction. Whereas transcendental equation is said to be such an equation that consists of transcendental functions. These equations are also called non-algebraic and will be infinite in nature.

4. What is meant by transcendental equations?

There should be a presence of a transcendental function in order to claim that an equation is a transcendental equation. This function can be of various types and can be provided as follows:

  1. Exponential

  2. Trigonometric

  3. Logarithmic

  4. Hyperbolic functions

  5. Reverse trigonometric

Two of the most important classes of transcendental functions are cylindrical and spherical functions. These sorts of functions are seen in problems of heat transfer which also includes the gamma and beta functions, Euler’s functions, hypergeometric, and degenerate hypergeometric functions. When solving problems related to heat transfer involves finding out the eigenvalues, in such sort of equations, there is a need for transcendental equations.


5. How do you solve a transcendental equation?

There are not many methods through which the transcendental equations can be solved, unlike algebraic equations where they can be obtained in a reduced form. However, there are lots of other ways through which these equations can be solved. Some of the methods that are used can be provided as follows:

  1. Newton’s Method: In this method, students need to assume that the function has a continuous derivative. This can be guessed to provide the following equation:

\[ x_{1} = x_{0} - \frac{f(x_{0})}{{f}'(x_{0})}\]

This method is used until an approximate solution is obtained.

  1. Ridder’s Method: This method uses a false position method which is combined with equations in order to produce a quadratic convergence to a real root.

  2. Secant Method: This method is quite similar to Newton’s method where a finite difference is used which is similar to the limit form of a derivation.

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