[Maths Class Notes] on Transcendental Function Pdf for Exam

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_{xrightarrow 0}frac{secx-1}{sinx}]

Solution: 

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

[lim_{^{2}xrightarrow 0}frac{secx-1}{sinx}]

[=lim_{xrightarrow 0}frac{secx-tanx}{cosx}]

[=lim_{xrightarrow 0}frac{1.0}{1}]

= 1