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---------- X-Sun-Data-Type: text X-Sun-Data-Description: text X-Sun-Data-Name: text X-Sun-Charset: iso-8859-1 X-Sun-Content-Lines: 32 Liebe Demugies, >Für eine Komplexizitätsanalyse benötige ich eine Größenabschätzung, v.a. von >2-er-Potenz-Fakultäten. > >Möglichst exakt. > >z.B. (2^64)! = ??? Da bietet sich natuerlich die Stirling Formel an, die Fakultaeten fuer grosse Zahlen gut approximiert: N! = N^N Sqrt[2Pi N] / E^N Davon kann man dann den Logarithmus nehmen. Auf diese Weise bekomme ich: Log[2^64] = 8*10^20 ------------------------------------------------------------------------- Eberhard von Kitzing Abteilung Zellphysiologie Max-Planck-Institut fuer Medizinische Forschung Jahnstr. 29, D69120 Heidelberg, FRG FAX : +49-6221-486 459 (work) Tel.: +49-6221-486 467 (work) Tel.: +49-6221-385 129 (home) internet: vkitzing@silia.mpimf-heidelberg.mpg.de http://sunny.mpimf-heidelberg.mpg.de/people/vkitzing/ ---------- X-Sun-Data-Type: Mathematica X-Sun-Data-Description: Mathematica X-Sun-Data-Name: STirling.nb X-Sun-Charset: us-ascii X-Sun-Content-Lines: 149 (*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 2460, 90]*) (*NotebookOutlinePosition[ 3096, 113]*) (* CellTagsIndexPosition[ 3052, 109]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Die Stirling Formel", "Title"], Cell[CellGroupData[{ Cell[BoxData[ \(Stirling[x_] = x^x\ Sqrt[2 Pi\ x]/E^x\)], "Input"], Cell[BoxData[ \(E\^\(-x\)\ \@\(2\ \[Pi]\)\ x\^\(1\/2 + x\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(LogStirling[x_] = Simplify[PowerExpand[Log[Stirling[x]]]]\)], "Input"], Cell[BoxData[ \(\(-x\) + x\ Log[x] + 1\/2\ Log[2\ \[Pi]\ x]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a = 2^64\)], "Input"], Cell[BoxData[ \(18446744073709551616\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(N[LogStirling[a]]\)], "Input"], Cell[BoxData[ \(7.99877009219260415`*^20\)], "Output"] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1152}, {0, 900}}, WindowSize->{520, 600}, WindowMargins->{{155, Automatic}, {Automatic, 79}} ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1731, 51, 36, 0, 107, "Title"], Cell[CellGroupData[{ Cell[1792, 55, 71, 1, 27, "Input"], Cell[1866, 58, 76, 1, 35, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[1979, 64, 90, 1, 27, "Input"], Cell[2072, 67, 77, 1, 44, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2186, 73, 41, 1, 27, "Input"], Cell[2230, 76, 54, 1, 27, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[2321, 82, 50, 1, 27, "Input"], Cell[2374, 85, 58, 1, 29, "Output"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************) |