Jensen's formula

In mathematics , Jensen's formula gives a formula for the integration of an analytical function over the edge of a circle. The formula is named after the Danish mathematician Johan Ludwig Jensen , who first described it in 1899.

It is of fundamental importance in the Nevanlinna theory (value distribution theory).

formula

Let be an analytic function and let its zeros in the circular area for a . Then applies ${\displaystyle f\colon \mathbb {C} \to \mathbb {C} }$ ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle D_{r}=\left\{z\in \mathbb {C} :|z|<r\right\}}$ ${\displaystyle r>0}$

${\displaystyle \log |f(0)|=\sum _{k=1}^{n}\log {\frac {|a_{k}|}{r}}+{\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\,d\theta .}$

If in has no zeros, one obtains the Gaussian mean value theorem for the harmonic function . ${\displaystyle f}$ ${\displaystyle D_{r}}$ ${\displaystyle \log |f(z)|}$

Example: polynomials

After the fundamental theorem of algebra , each can polynomial over disassemble as ${\displaystyle \mathbb {C} }$

${\displaystyle p(z)=a\prod _{i=1}^{d}(z-\alpha _{i})}$.

From Jensen's formula it then follows with : ${\displaystyle r=1}$

${\displaystyle \log |a|+\sum _{i=1}^{d}\,\log(\max(|\alpha _{i}|,1))={\frac {1}{2\pi }}\int _{0}^{2\pi }\log |p(e^{i\theta })|\,d\theta .}$

Example: can be dismantled as with . Because of this it follows ${\displaystyle p(z)=z^{2}-z+1}$ ${\displaystyle p(z)=(z-\alpha _{1})(z-\alpha _{2})}$ ${\displaystyle \alpha _{1,2}={\tfrac {1}{2}}\pm {\tfrac {\sqrt {3}}{2}}i}$ ${\displaystyle |\alpha _{1}|=|\alpha _{2}|=1}$

${\displaystyle \int _{0}^{2\pi }\log |p(e^{i\theta })|\,d\theta =0}$.

literature

• J. Jensen: Sur un nouvel et important théorème de la théorie des fonctions. In: Acta Mathematica. (Springer Netherlands) 22, 1899, pp. 359-364. (French)
• P. Borwein, T. Erdélyi: Jensen's Formula. §4.2.E.10c In: Polynomials and Polynomial Inequalities. Springer-Verlag, New York 1995, ISBN 0-387-94509-1 , p. 187.
• SG Krantz: Jensen's Formula. §9.1.2 In: Handbook of Complex Variables. Birkhäuser, Boston MA 1999, ISBN 3-7643-4011-8 , pp. 117-118.

Web links

• Eric W. Weisstein : Jensen's Formula . In: MathWorld (English).