Transcendental function

A transcendental function is an analytic function that is not algebraic . The simplest examples of transcendental functions are exponential function , trigonometric functions , logarithmic function .

If transcendental functions are considered as functions of a complex variable, then their characteristic feature is the presence of at least one feature different from poles and branch points of finite order.

For example, $e^{z}$ ${\ displaystyle e ^ {z}}$ $e ^ {z}$ ; $\cos z$ ${\ displaystyle \ cos z}$ ${\ displaystyle \ cos z}$ and $\sin z$ ${\ displaystyle \ sin z}$ ${\ displaystyle \ sin z}$ have a significant singular point $z=\infty$ ${\ displaystyle z = \ infty}$ $z = \ infty$ (Where $\infty$ ${\ displaystyle \ infty}$ $\ infty$ denotes the vertex of the Riemann sphere - an infinitely distant point of the complex plane), $\ln z$ ${\ displaystyle \ ln z}$ ${\ displaystyle \ ln z}$ - branch points of infinite order for $z=0$ ${\ displaystyle z = 0}$ ${\ displaystyle z = 0}$ and $z=\infty$ ${\ displaystyle z = \ infty}$ $z = \ infty$ .

The foundations of the general theory of transcendental functions are given by the theory of analytic functions. Special transcendental functions are studied in the relevant disciplines (the theory of hypergeometric , elliptic , Bessel functions, etc.).

Transcendental function

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