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Concept
Lindelöf's theorem
Summary
In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip must remain bounded on the whole strip. The result is useful in the study of the Riemann zeta function, and is a special case of the Phragmén–Lindelöf principle. Also, see Hadamard three-lines theorem.
Let be a half-strip in the complex plane:
Suppose that is holomorphic (i.e. analytic) on and that there are constants
, and such that
and
Then is bounded by on all of :
Fix a point inside . Choose , an integer and large enough such that
Applying maximum modulus principle to the function and
the rectangular area we obtain , that is, . Letting yields
as required.
Official source
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