What Is a Transcendental Function Definition Types Properties and Solved Examples
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 Explained with Definition and Key Concepts
1. What is a transcendental function?
A transcendental function is a function that cannot be expressed as a finite combination of algebraic operations (addition, subtraction, multiplication, division, and roots).
In other words, transcendental functions are not solutions of polynomial equations with finite terms.
Common examples include:
- Exponential functions such as ex
- Logarithmic functions such as ln x
- Trigonometric functions such as sin x and cos x
2. What is the difference between algebraic and transcendental functions?
The key difference is that an algebraic function satisfies a polynomial equation, while a transcendental function does not.
- Algebraic functions: Built using polynomials and roots (e.g., x² + 3x, √x).
- Transcendental functions: Cannot be written as a finite polynomial expression (e.g., ex, ln x, sin x).
3. Is e^x a transcendental function?
Yes, ex is a transcendental function because it cannot be expressed as a finite polynomial or algebraic expression.
The exponential function:
- Has derivative equal to itself: d/dx (ex) = ex
- Is defined for all real numbers
- Appears in growth and decay models
4. Are trigonometric functions transcendental?
Yes, trigonometric functions like sin x, cos x, and tan x are transcendental functions.
They are transcendental because:
- They cannot be expressed as finite algebraic expressions
- They have infinite power series expansions
- They are not solutions of polynomial equations with finite terms
5. What is an example of a transcendental equation?
A transcendental equation is an equation that contains at least one transcendental function.
Example:
- ex = 3
- Take natural logarithm on both sides.
- x = ln 3
6. How do you solve a transcendental equation?
To solve a transcendental equation, you typically use logarithms, graphical methods, or numerical methods.
Common approaches:
- Logarithms: For equations like ex = 5, take ln on both sides.
- Graphical method: Plot both sides and find intersection points.
- Numerical methods: Use Newton-Raphson or iteration for complex equations.
7. Why are transcendental functions important in calculus?
Transcendental functions are important in calculus because they model growth, oscillation, and continuous change.
They are widely used in:
- Differentiation and integration (e.g., derivative of ex is ex)
- Differential equations
- Physics and engineering models
8. Is ln x a transcendental function?
Yes, ln x (natural logarithm) is a transcendental function because it cannot be expressed as a finite algebraic expression.
Key properties:
- Defined for x > 0
- Derivative is 1/x
- Inverse of the exponential function ex
9. What is the power series of a transcendental function?
Many transcendental functions can be represented by an infinite power series.
For example:
- ex = 1 + x + x²/2! + x³/3! + ...
- sin x = x − x³/3! + x⁵/5! − ...
10. Can a transcendental function be a polynomial?
No, a transcendental function cannot be a polynomial because polynomials are algebraic functions by definition.
A polynomial:
- Has finite terms
- Uses only powers of x with constants
- Satisfies a polynomial equation
