New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | MEDIUM | LARGE Below is the OCR-scanned text from this page: 6. Gamma Function and Related Functions Ma thema tical Properties 6.1. Gamma (Factorial) Function Euler's Integral 6.1.1 r ( z ) = tz-'e-' d t (92>0) Lrn =pirn tz-le-kt d t ( 9 Z > O , Wk>O) Euler's Formula 6.1.2 Y t n! nz r (2) = lim (zf0,-1,-2,. . .) n+rn z ( z + ~ ) . . . (z+n> Euler's Infinite Product 1 1 1 1 y= lim I+-+-+-+. . . +--In m] m+-[ 2 3 4 m. =.57721 56649. . . Y is known as Euler's constant and is given to 25 decimal places in chapter 1. r(z) is single valued and analytic over the entire complex plane, save for the points z=-n(n=O, 1, 2, . . . ) where it possesses simple poles with residue (- l)"/n!. Its reciprocal i/r (z) is an entire function possessing simple zeros at the points z=-n(%=O, 1, 2, . . .). Hankel's Contour Integral FIGURE 6.1. Gamma function. * , y = r ( ~ ) , - - - - , Y=Ww 1.5.9-13. . . (4n-3) 4" rca, 6.1.10 r (n+ 2) = The path of integration C starts a t + m on the real axis, circles the origin in the counterclockwise direction and returns to the starting point. Factorial and II Notations 6.1.5 D(Z) = z!= r(z+ 1) Integer Values 6.1.6 6.1.7 r(n+1)=1.2.3 . . . (n-l)n=n! 1.4.7.10. . . (3n-2) 3" r ( 3 ) 6.1.11 r (n+ 3 ) = r(3) 2.5-8.11 . . . (3%-1) 3" 6.1.13 r(n+\$)= (n=O, 1, 2, . . .) ' L o = 1 lim ~ z+n r(-z)- (-%-I)! r(#)=1.35411 79394.. . 3-7.11.15. . . (4%-1) 4" r (9) 6.1.14 r(n+i) = Fractional Values 6.1.8 r(4) = 2 L m e-12dt=r3=1.77245 38509 . . . =(-\$)! The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States. ©2000 ConvertIt.com, Inc. All rights reserved. Terms of Use.