Lindelöf's theorem

Lindelöf's theorem

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.

tatement of the theorem

Let Ω be a half-strip in the complex plane:

:Omega = { z in mathbb{C} | x_{1} leq mathrm{Re} (z) leq x_{2} mbox{ and } mathrm{Im} (z) geq y_{0} } subsetneq mathbb{C}.

Suppose that "f" is holomorphic (i.e. analytic) on Ω and that there are constants "M", "A" and "B" such that

:| f(z) | leq M mbox{ for all } z in partial Omega

and

:frac{y^{A leq B mbox{ for all } x + i y in Omega.

Then "f" is bounded by "M" on all of Ω:

:| f(z) | leq M mbox{ for all } z in Omega.

References

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