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Problem No.4 Based on Transcendental Functions | Ekeeda.com

55 views · Published 1 December 2016 · 4:16 · Indexed 22 September 2026

Channel: Ekeeda · 2016 · Education

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Transcendental function:
A transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. (The polynomials sometimes must have rational coefficients. 
A transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e.
A transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.[1][2] (The polynomials sometimes must have rational coefficients.
a transcendental function "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction.
The most familiar transcendental functions are the logarithm, the exponential (with any non-trivial base), the trigonometric, and the hyperbolic functions, and the inverses of all of these. Less familiar are the special functions of analysis, such as the gamma, elliptic, and zeta functions, all of which are transcendental. The generalized hypergeometric and Bessel functions are transcendental in general, but algebraic for some special parameter values.
A function that is not transcendental is algebraic. Simple examples of algebraic functions are the rational functions and the square root function, but in general, algebraic functions cannot be defined as finite formulas of the elementary functions.
The indefinite integral of many algebraic functions is transcendental. For example, the logarithm function arose from the reciprocal function in an effort to find the area of a hyperbolic sector.
Differential algebra examines how integration frequently creates functions that are algebraically independent of some class, such as when one takes polynomials with trigonometric functions as variables.
Most familiar transcendental functions, including the special functions of mathematical physics, are solutions of algebraic differential equations. Those that are not, such as the gamma and the zeta functions, are called transcendentally transcendental or hypertranscendental functions.



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