Modal analysis using FEM - Revision history

134.223.230.155: /* After FEA eigensystems */

2014-12-19T19:23:01Z

After FEA eigensystems

en>Wavelength: inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

2014-02-28T01:50:37Z

inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

← Older revision		Revision as of 03:50, 28 February 2014
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	~~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 ''&fnof;''~~ 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]~~]

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q3780658

2013-03-02T05:57:12Z

Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q3780658

← Older revision		Revision as of 07:57, 2 March 2013
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	~~The writer~~ is ~~called Araceli Gulledge~~. ~~Meter studying~~ is ~~exactly where my primary income arrives from but soon I'll be~~ on ~~my personal~~. ~~Kansas~~ is ~~exactly where her house~~ is ~~but she needs to transfer simply because~~ of ~~her family members~~. ~~To keep birds~~ is ~~one~~ of ~~the issues he loves most~~.<br><br>~~Also visit my weblog~~ - ~~[http:~~//~~Indianapolisfaith~~.~~org~~/~~UserProfile~~/~~tabid~~/~~320~~/~~userId~~/~~242895~~/~~Default~~.~~aspx extended vehicle warranty~~]		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]].

			==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 ''&fnof;'' 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]]

en>Dr eng x: /* FEA eigensystems */ expansion is needed

2012-05-23T18:05:45Z

FEA eigensystems: expansion is needed

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The writer is called Araceli Gulledge. Meter studying is exactly where my primary income arrives from but soon I'll be on my personal. Kansas is exactly where her house is but she needs to transfer simply because of her family members. To keep birds is one of the issues he loves most.<br><br>Also visit my weblog - [http://Indianapolisfaith.org/UserProfile/tabid/320/userId/242895/Default.aspx extended vehicle warranty]