Entire function

Definition

In the mathematical analysis and, in particular, in the theory of functions of complex variable, The entire function is function that is holomorphic in the whole complex plane.

Examples

Examples of entire functions are the polynomials and the exponentials. All sums, products and compositions of these functions also are entire functions.

All the derivatives and some of integrals of entired funcitons, for example erf, Si, $J_{0}$, also are entired functions.

Every entire function can be represented as a power series or Tailor expansion which converges everywhere.

In general, neither series nor limit of a sequence of entire funcitons needs to be an entire function.

Inverse of an entire function has no need to be entire function.

Examples of non-entire functions: rational function $~f(z)={\frac {a+bx}{c+x}}~$ at any complex $~a~$, $~b~$, $~c~$ , square root, logarithm, funciton Gamma, tetration.

Properties

Infinitness

Liouville's theorem establishes an important property of entire functions — an entire function which is bounded must be constant. This property can be used for an elegant proof of the fundamental theorem of algebra. Picard's little theorem is a considerable strengthening of Liouville's theorem: a non-constant entire function takes on every complex number as value, except possibly one. The latter exception is illustrated by the exponential function, which never takes on the value 0.

Cauchi integral

Entire function $~f~$, at any complex $~z~$ and at any contour C evolving point $z$ just once, can be expressed with Cauchi theorem

$f(x)={\frac {1}{2\pi {\rm {i}}}}\oint _{\mathbf {C} }{\frac {f(t)}{t-z}}{\rm {d}}t$

See also

• Cauchi formula
• Tailor series