By Gradshteyn I., Ryzhik I.

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**Extra resources for Table of Integrals Series and Product**

**Example text**

In this case, the error (in absolute value) does not exceed the absolute value of the first of the discarded terms (cf. ). Asymptotic series have many properties that are analogous to the properties of convergent series and, for that reason, they play a significant role in analysis. The asymptotic expansion of a function is denoted as follows: f (z ) » 1 P An z ¡n : n=0 23 The definition of an asymptotic expansion. The divergent series 1 P n=0 An zn is called the asymptotic expansion of a function f (z ) in a given region of values of arg z if the expression Rn (z ) = z n [f (z ) ¡ Sn (z )]; where Sn (z ) = lim Rn (z ) = 0 for fixed n.

All rights reserved. 5) 3: etg x = 1 + x + x2 3x3 9x4 37x5 + + + + ... 2! 3! 4! 5! 216 1: earcsin x = 1 + x + x2 2x3 5x4 + + + ... 2! 3! 4! 7) 2: earctg x = 1 + x + x2 x3 7x4 ¡ ¡ + ... 2! 3! 4! 217 1: ¼ 1 P e¼x + e¡¼x 1 1 = + 2x (cf. 421 2: e¼x 1 P 1 2¼ 1 = + 2x (¡1)k 2 ¡ ¼x ¡e x x + k2 k=1 (cf. 221 1: ax = ex ln a : 2: aloga x = a logx a = x: 1 Printed from Table of Integrals, Series, and Products CD-ROM. Copyright © 1996 by Academic Press, Inc. All rights reserved. 30 Introduction The trigonometric and hyperbolic sines are related by the identities sh x = 1 sin (ix); i sin x = 1 sh (ix): i Printed from Table of Integrals, Series, and Products CD-ROM.

Copyright © 1996 by Academic Press, Inc. All rights reserved. 430 If f(x)=F(y) and y='(x), then 1: dn U1 0 U2 00 U3 000 Un (n) f (x) = F (y ) + F (y ) + F (y ) + . . + F (y ) ; dxn 1! 2! 3! n! where Uk = dn k k dn k(k ¡ 1) 2 dn k¡2 dn y y ¡ y n y k ¡1 + y y ¡ . . + (¡1)k¡1 ky k¡1 n : dxn 1! dx 2! dxn dx AD (7361) GO 2: P dn n! h! . k! dy m Here, the symbol P µ y0 1! ¶i µ y 00 2! ¶j µ y 000 3! ¶h ¢¢¢ µ y (l) l! ¶k ; indicates summation over all solutions in non negative integers of the equation i + 2j + 3h + .