|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''Lindelöf's theorem''' is a result in [[complex analysis]] named after the [[Finland|Finnish]] [[mathematician]] [[Ernst Leonard Lindelöf]]. It states that a [[holomorphic function]] on a half-strip in the [[complex plane]] that is [[bounded function|bounded]] on the [[boundary (topology)|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]].
| | Yoshiko is her name but she doesn't like when individuals use her full title. I am a production and distribution officer. One of my favorite hobbies is camping and now I'm trying to make money with it. Alabama is where he and his spouse live and he has every thing that he needs there.<br><br>Also visit my weblog ... [http://bikedance.com/blogs/post/29704 bikedance.com] |
| | |
| ==Statement of the theorem==
| |
| Let Ω be a half-strip in the complex plane:
| |
| | |
| :<math>\Omega = \{ z \in \mathbb{C} | x_1 \leq \mathrm{Re} (z) \leq x_2 \text{ and } \mathrm{Im} (z) \geq y_0 \} \subsetneq \mathbb{C}. \, </math>
| |
| | |
| Suppose that ''ƒ'' is [[holomorphic]] (i.e. [[analytic function|analytic]]) on Ω and that there are constants ''M'', ''A'' and ''B'' such that
| |
| | |
| :<math>| f(z) | \leq M \text{ for all } z \in \partial \Omega \,</math>
| |
| | |
| and
| |
| | |
| :<math>\frac{| f (x + i y) |}{y^{A}} \leq B \text{ for all } x + i y \in \Omega. \,</math>
| |
| | |
| Then ''f'' is bounded by ''M'' on all of Ω:
| |
| | |
| :<math>| f(z) | \leq M \text{ for all } z \in \Omega. \,</math>
| |
| | |
| ==Proof==
| |
| Fix a point <math>\xi=\sigma+i\tau</math> inside <math>\Omega</math>. Choose <math>\lambda>-y_0</math>, an integer <math>N>A</math> and <math>y_1>\tau</math> large enough such that
| |
| <math>\frac {By_1^A}{(y_1 + \lambda)^N}\le \frac {M}{(y_0+\lambda)^N}</math>. Applying [[maximum modulus principle]] to the function <math>g(z)=\frac {f(z)}{(z+i\lambda)^N}</math> and
| |
| the rectangular area <math>\{z \in \mathbb{C} | x_1 \leq \mathrm{Re} (z) \leq x_2 \text{ and } y_0 \leq \mathrm{Im} (z) \leq y_1 \}</math> we obtain <math>|g(\xi)|\le \frac{M}{(y_0+\lambda)^N}</math>, that is, <math>|f(\xi)|\le M\left(\frac{|\xi + \lambda|}{y_0+\lambda}\right)^N</math>. Letting <math>\lambda \rightarrow +\infty</math> yields
| |
| <math>|f(\xi)| \le M</math> as required.
| |
| | |
| ==References==
| |
| *{{cite book|author=Edwards, H.M.|authorlink=Harold Edwards (mathematician)|title=Riemann's Zeta Function|publisher=Dover|location=New York, NY|year=2001|isbn=0-486-41740-9}}
| |
| | |
| {{DEFAULTSORT:Lindelof's theorem}}
| |
| [[Category:Theorems in complex analysis]]
| |
Yoshiko is her name but she doesn't like when individuals use her full title. I am a production and distribution officer. One of my favorite hobbies is camping and now I'm trying to make money with it. Alabama is where he and his spouse live and he has every thing that he needs there.
Also visit my weblog ... bikedance.com