Transcendental Functions

The transcendental function is a mathematical expression that is basically expressed in non-algebraic form, due to which it is also called a non-algebraic equation. The basic representation of a transcendental function is through exponential function,  trigonometric metrical function, logarithm, function, etc. Transcendental function calculus includes integration, derivation, and limit evaluation of various equations. Aleksandr Osipovich Gelfond is a mathematician who conducted a study on this non-algebraic equation which cannot be expressed in the form of multiplication, division, raised to the power, or any other algebraic function. He also postulated several theories related to these transcendental maths functions, like the theory of interpolation and the advanced transcendental. Number theory.

Transcendental Maths Functions

A transcendental function is a mathematical equation or function that cannot be expressed in the form of algebraic equations, i.e., a transcendental function cannot be expressed in the form of polynomial expression, which involves rational coefficients. In simple terms, the transcendental function transits beyond the algebraic expressions, which cannot be represented in the expression of subtraction, multiplication, raised to a power, etc. It is also called a non-algebraic equation. The transcendental function can be expressed in the form of exponential function or trigonometric functions.

Transcendental Numbers

Transcendental numbers are numbers that cannot be expressed in an algebraic form or don’t have a definite solution. For example, x2-4=0, then the following function is not a transcendental number because there is a clear solution for the function, i.e., x=4=2.

Therefore the equation that is in the form of algebraic expressions is not termed as transcendental numbers. Whereas mathematical expressions like e and π are termed transcendental numbers.

It is considered that every transcendental function is an irrational number, whereas vice versa is not applicable. All irrational numbers are not termed transcendental numbers.

Algebraic and Transcendental Functions

Algebraic and transcendental functions are distinctive to each other in terms of mathematical representation and functions. An algebraic equation or function can be defined as a mathematical function that is considered the root of any polynomial expression with a rational coefficient. The algebraic function is represented in terms of subtraction multiplication, fractional power, division, etc. An Algebraic equation can be expressed in a function form of y=f(x) is the basic pattern for any mathematics polynomial expression.

Considering an equation,

P0(x)yn+Pq(x)+….. Pn(x)=0

Here, the function P0(x), P1(x), ….. , Pn(x)=0 is in the form of a polynomial, expressed in x. Therefore, every rational and irrational function is expressed as an algebraic function.

On the other hand, a transcendental function is those mathematical expressions that cannot be represented in terms of an algebraic function. They are expressed in the form of trigonometrical fiction, logarithmic function, hyperbolic function, i.e., logx , sinx , ex, as well as in terms of an inverse function. An inverse function can be expressed as the relationship,

f(x)=y, which can be termed as an inverse function if c(y)=x.

Here, c is said to be the inverse of f, and this relationship is termed as transcendental function.

Transcendental Function Calculus

Aleksandr Osipovich Gelfond is a Russian mathematician who studied algebraic functions and the numbers that can be termed in the form of non-algebraic functions. He came up with significant methods for exploring transcendental numbers. The theory of interpolation and the advanced transcendental number theory were postulated by Aleksandr to explain the transcendental function calculus. The representation of the transcendental function calculus expresses the continuous change in the equation. There are two groups of calculus that deal with transcendental functions. They are differential calculus and integral calculus. The differential calculus is used to evaluate the continuous change in the equation, and the integral calculus is used to determine the areas, volumes, and several other mathematical facts.

Conclusion

The transcendental functions are the mathematical expressions that are not able to satisfy a polynomial equation, i.e., they cannot be represented in the form of algebraic equations like multiplication, raised to a power, division, etc. The transcendental function is represented by e, , etc., which is the expression for the exponential function trigonometric function. It is also represented as logarithm function, hyperbolic function, etc. Transcendental function calculus representation expresses the continuous change in the equation as well as the evaluation of various mathematical expressions. It is considered that every transcendental function is an irrational expression, whereas vice versa is not applicable. To understand the concept of transcendental function better, one is required to have a basic knowledge of integral transform, trigonometric functions, and inverse function.