a transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves roots of polynomials, in contrast to an algebraic function, which does satisfy such an equation.e. j. townsend, functions of a complex variable, bibliolife, llc, (2009). (the polynomials are sometimes required to have rational coefficients.) in other words, a transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction. examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. ==definition== formally, an analytic function &fnof;(z) of one real or complex variable z is transcendental if it is algebraically independent of that variable.m. waldschmidt, diophantine approximation on linear algebraic groups, springer (2000). this can be extended to functions of several variables. ==examples== the following functions are transcendental: :f_1(x) = x^\pi \ :f_2(x) = c^x, \ c \ne 0, 1 :f_3(x) = x^{x} = {{^2}x} \ :f_4(x) = x^{\frac{1}{x}} \ :f_5(x) = \log_c x, \ c \ne 0, 1 :f_6(x) = \sin{x} note that in particular for Æ2 if we set c equal to e, the base of the natural logarithm, then we get that ex is a transcendental function. similarly, if we set c equal to e in Æ5, then we get that ln(x), the natural logarithm, is a transcendental function. for more information on the second notation of Æ3, see tetration. ==algebraic and transcendental functions==  the logarithm and the exponential function are examples of transcendental functions. transcendental function is a term often used to describe the trigonometric functions (sine, cosine, tangent, their reciprocals cotangent, secant, and cosecant, the now little-used versine, haversine, and coversine, their analogs the hyperbolic functions and so forth). a function that is not transcendental is said to be algebraic. examples of algebraic functions are rational functions and the square root function. the operation of taking the indefinite integral of an algebraic function is a source of transcendental functions. for example, the logarithm function arose from the reciprocal function in an effort to find the area of a hyperbolic sector. thus the hyperbolic angle and the hyperbolic functions sinh, cosh, and tanh are all transcendental. differential algebra examines how integration frequently creates functions that are algebraically 
Analytic_functions Functions_and_mappings Meromorphic_functions Special_functions Types_of_functions