Formulate of Jensen

The formula of Jensen (according to the mathematician Johan Jensen) is a result of Analyze complexes which describes the behavior of a analytical Fonction on a circle by brings back to the Module S Zero S of this function. It is of an invaluable help for the study of the whole functions.

The statement is the following:

Either F is an analytical function on an area of the Plan complex containing a disc closed D of center 0 and or $a\_1,\; a\_2,\; \backslash \; dowries,\; a\_n$ the zeros of F in D , taken into account their multiplicity. If $f\; (0)\; \backslash \; 0$, then

$\backslash \; log\; |F\; (0)|\; =\; -\; \backslash \; sum\_\; \{k=1\}\; ^n\; \backslash \; log\; \backslash \; left\; (\backslash \; frac\; \{R\}\; \backslash \; right)\; +\; \backslash \; frac\; \{1\}\; \{2\; \backslash \; pi\}\; \backslash \; int\_0^\; \{2\; \backslash \; pi\}\; \backslash \; log|F\; (re^\; \{I\; \backslash \; theta\})|D\; \backslash \; theta.$

This formula establishes a bond between the modules of the zero contents in a disc theorem of Poisson-Jensen .

References

• Complex Analysis , John Mr. Howie, Springer-Verlag

• Complex Analysis , L.V. Ahlfors, McGraw-Hill, 1979, ISBN 0-07-000657-1

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