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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: ELEMENTARY TRANSCENDENTAL FUNCTIONS 81 4.4.39 26 Arctan z=ka++ arctan where k is an integer or zero and a = + [ (x + 1) 2 + ?p] + + 3 [ (x - 1) + yZ] P=h [(x+1)2+y2]+-~ [(x- 1)"3/2]* Series Expansions 4.4.40 4.4.41 arcsin (1-2)=?-(22)* 1 + ~ 1 . 3 - 5 . . . (2k-1) 2 [ k = i 22k(2k+l)k! (I z I landz2#-1) - 7 r 1 +--- 1 1 _--- 2 z 323 525 _- - Z 2 -+""( 22 z2 J+ . . .] l+z"l+Jl+zz 3 . 5 1\$.z (z2\$ - 1) Continued Fractions z 2' 42' 92' 1 6 9 4.4.43 arctan z=- I+ ZT 5+7f 9\$", ( z in the cut plane of Figure 4.4.) arcsin z- z 1 . 2 ~ ~ 1 . 2 ~ ~ 3 . 4 2 ~ 3 . 4 ~ ~ 1- 3- 5- 7- 9- . * . ( z in the cut plane of Figure 4.4.) 4.4.44 Polynomial Approximations Q 4.4.45 O l x l l 7r arcsin x=s- ( 1 - x) t (ao +alx + a2x2 + a3x3) + e (x) I e (x) I 5 5 x 1 0-5 ao= 1.57072 88 uZ= .07426 10 a1=-.21211 44 a3=-.01872 93 Q The approximations 4.4.45 to 4.4.47 are from C. Hast- ings, Jr., Approximations for digital computers. Princeton Univ. Press, Princeton, N.J., 1955 (with permission). 4.4.46 O