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 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 | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: GAMMA FUNCTION AND RELATED FUNCTIONS 263 6.5.30 Definite Integrals 6.5.36 Suppose li',(a,z)=u,,,(a,z)+ . . . is the re- mnitider after n tcrms in this series. Then if a,z 6.5.37 Continued Fraction 6.6.3 Iz(a,b)=l-I~-z(b,a) Relation to Binomial Expansion Asymptotic Expansions 6.5.32 6.7. Use and Extension of the Tables Example 1. Compute r(6.38) to 8s. Using tile rwitrrcticc rclntiori 6.1.26 and Table 6.1 we ilavc, r(6.38)=[(5.:38)(4.:58)(:3.38)(2.:38)(1.38)]r(1.38) =2:32.43671. Example 2. Compute ln r(56.38), 11si1lg Table 6.4 and liticnr itittrpolation iti-f2. I\-(> liavc (z>O,l-al< a) 6.6. Incomplete Beta Function The error of linear interpolation in the table of tlic fuiictionjz is smaller tlian 10-7 in this region. HCIlCC, fz(5&38) =.92041 67 atid In ~ ( 5 6 . 3 8 ) ~ 169.85497 42. Dircct interpolation in Table 6.4 of log,, r(n) eliminates the neccssity of employing logarithms. Howwer, tltc error of linear interpolation is .002 so that log,, r ( n ) is obtained with a relative error of 10-5. / 6.6.1 B, (a, b) = J t = - (1 - t ) - 'clt 6.6.2 (a,b) =BZ(a,b) lB('Jb) For statistical applications, see 26.5. Symmetry 6.6.4 I p ( a J n - a + l ) = 5 6) p3(l--p)"-j ] = a a t i d sign I?, (a,z) =sign u,+, (a,z). 1 For binomial distribution, see 26.1. 6.5.35 Recurrence Formulas (a++ a) 6.6.5 Iz(~,b)==xIZ(~- l,b)+ ( l - ~ ) I ~ ( a , b - 1 ) * 6.6.6 (a+ b-m)L(a,b) =a ( 1 - 2) 1, (a + 1 ,b - 1 ) + bIZ (a, b + 1 ) 6.6.7 ( ~ + b ) l , ( ~ , b ) =u~,(u+ 1,b) +bIz(a,b+ 1) Relation to Hypergeometric Function In r(56.38) = (56.38-+) In (56.38) - (56.38) *See page 11. +fi (56.3 8) 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.