Jensen's formula

Jensen's formula relates the circular mean of the logarithm of a holomorphic function's modulus on a circle to its value at the center and the zeros inside the circle.

Holomorphic function: A complex-valued function that is complex differentiable throughout an open set. Jensen's formula applies to holomorphic functions on a disk containing the circle and its interior.
Complex analysis: The mathematical study of functions of complex variables and their analytic properties. Jensen's formula is a central identity for holomorphic functions in this subject.
Jensen's inequality for zeros: An upper bound on the number of zeros in a smaller disk derived from Jensen's formula and boundary bounds. It converts the formula's weighted zero sum into a count of zeros.
Johan Jensen: A Danish mathematician known for work in complex analysis, inequalities, and the theory of entire functions. Jensen introduced the formula in his work on entire functions.
Circular mean: The average of a function's values around a circle, computed with respect to angular measure. The boundary term averages the logarithmic modulus over the circle.
Harmonic function: A twice-differentiable function whose Laplacian is zero on its domain. Away from zeros, the logarithmic modulus of a holomorphic function is harmonic.
Blaschke condition: The summability condition on zeros required for a bounded holomorphic function in the unit disk to have a Blaschke product factor. Jensen's formula yields the necessary summability of zero locations for bounded functions.
Entire function: A function holomorphic on the whole complex plane. Jensen developed the identity as a tool for studying zeros and growth of entire functions.
Logarithmic modulus: The real-valued function obtained by taking the logarithm of a complex function's absolute value. Its boundary mean is balanced against the contributions from interior zeros.
Multiplicity of a zero: The order of the first nonzero term in a holomorphic function's local power series at a zero. Repeated zeros enter the formula with their full multiplicities.

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Carlson's theoremRelated: Zero-counting estimates help explain why the allowed growth cannot sustain all the integer zeros.