# Whole function

In function theory , a whole function is a function that is holomorphic (i.e. analytical ) in the entire complex number plane . Typical examples of whole functions are polynomials or the exponential function as well as sums, products and combinations thereof, such as the trigonometric functions and the hyperbolic functions . ${\ displaystyle \ mathbb {C}}$

## properties

Every whole function can be represented as an everywhere converging power series around any center. Neither the logarithm nor the root function are whole.

A whole function can have an isolated singularity , in particular even an essential singularity in the complex point at infinity (and only there).

An important property of whole functions is Liouville's theorem : Bounded whole functions are constant . This is a very elegant way to prove the fundamental theorem of algebra . The small set of Picard is a substantial tightening of the set of Liouville: A non-constant entire function accepts all the values of the complex plane, except for possibly a. The latter exception illustrates, for example, the exponential function , which does not take the value 0.

## Further examples

The Airy function (here the real part) is a whole function.${\ displaystyle \ operatorname {Bi} (x + iy)}$
• the reciprocal of the gamma function ${\ displaystyle 1 / \ Gamma (z)}$
• the error function ${\ displaystyle \ operatorname {erf} (z)}$
• the integral sine ${\ displaystyle \ operatorname {Si} (z)}$
• the Airy functions and${\ displaystyle \ operatorname {Ai} (z)}$${\ displaystyle \ operatorname {Bi} (z)}$
• the Fresnel integrals and${\ displaystyle S (z)}$${\ displaystyle C (z)}$
• the Riemann Xi function ${\ displaystyle \ xi (z)}$
• the Bessel functions of the first kind for whole numbers${\ displaystyle J_ {n} (z)}$${\ displaystyle n}$
• the Struve functions for integers${\ displaystyle H_ {n} (z)}$${\ displaystyle n> -2}$
• the greatest common divisor with respect to a natural number in the generalized form ${\ displaystyle n}$
• ${\ displaystyle \ textstyle \ operatorname {ggT} (n, z) = \ sum \ limits _ {m = 1} ^ {n} e ^ {2 \ pi i {\ frac {m} {n}} z} \ sum \ limits _ {d | n} {\ frac {c_ {d} (m)} {d}} \ quad {\ text {with}} \ quad c_ {d} (m) = \! \! \! \! \ sum \ limits _ {k = 1 \ atop \ operatorname {ggT} (k, d) = 1} ^ {d} \! \! \! \! e ^ {2 \ pi i {\ frac {k }{dm}}$( Ramanujan sum )

## literature

• Klaus Jänich : Function Theory . An introduction . 6th edition. Springer-Verlag, Berlin / Heidelberg 2004
• Reinhold Remmert : Analysis I . 3. Edition. Springer-Verlag, Berlin Heidelberg 1992
• Eberhard Freitag , Rolf Busam: Function Theory 1 . 3. Edition. Springer-Verlag, Berlin / Heidelberg 2000