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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: 5. Exponential Integral and Related Functions Mathematical Properties 5.1. Exponential Integral Definitions m e - l 5.1.1 E , ( z ) = s 2 t' dt (larg zlO) 5.1.2 Ei(x)=-Lm - dt=- e-' J - = t J - = t 5.1.4 5.1.5 ffn ( 2 ) = lm tne-zrdt (n=O, 1 , 2 , . . .; -5%z>O) 5.1.6 &(z)= tne-zrdt (n=O, 1, 2, . . . ) S I I In 5.1.1 it is assumed that the path of integration excludes the origin and does not cross the negative real axis. Analvtic continuation of the functions in 5.1.1, 5.1.2, a i d 5.1.4 for n>O yields multi-valued funci tions with branch points at z=O and ~ = m . ~ They are single-valued functions in the z-plane cut along the negative real axis.' The function li(z), the logarithmic integral, has an additional branch point at z= 1. Interrelations 5.1.7 El (--z f i0) = -Ei(-z) ~ i r , -Ei(z) = \$[El (- z+iO) +El (- 2- io) ] (-z>O) a Some authors [5.14], [5.16] use the entire function l ( l - e - ' ) d t / t as the basic function and denote it by Ein(z). We have Ein(z)=El(z)+ln z f r . * Various authors define the integral J: (e'/t)dt in the z-plane cut along the positive real axis and denote italso by Ei(z). For z=z>O additional notations such as Ei(z). (e.g., in [5.10], [5.25]), E*(z) (in [5.2]), Ei*(z) (in [5.6]) are then used to designate the principal value of the integral. Correspondingly, El(%) is often denoted by -Ei(-z). 228 Explicit Expressions for a,(z) and Pn(z) 5.1.8 a,(z)=n!z-n-le-z (l+z+z+ 22 . . . +a) 2" pn(z)=n!z-n-l{ez [1--z+z-. 22 . . +(-l)"aI 2" 5.1.9 Y A FIGURE 5.1. y=Ei(z) and y=El(-z). i/ FIGURE 5.2. y = E n ( X ) n=O, 1,2, 3, 5, 10 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.