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 ${\displaystyle n}$
  • ${\displaystyle \textstyle \operatorname {ggT} (n,z)=\sum \limits _{m=1}^{n}e^{2\pi i{\frac {m}{n}}z}\sum \limits _{d|n}{\frac {c_{d}(m)}{d}}\quad {\text{mit}}\quad c_{d}(m)=\!\!\!\!\sum \limits _{k=1 \atop \operatorname {ggT} (k,d)=1}^{d}\!\!\!\!e^{2\pi i{\frac {k}{d}}m}}$( 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

Web links

• Eric W. Weisstein : Entire Function . In: MathWorld (English).

Individual evidence

1. ^ Wolfgang Schramm: The Fourier transform of functions of the greatest common divisor . In: Integers - The Electronic Journal of Combinatorial Number Theory , 8, 2008, A50 ( abstract )