Interpreting a paper: ei and erf

CRGreathouse · 2009-05-13, 23:01

From a paper I've been reading:

Quote:

This class includes, in addition to the elementary functions, a number of well-known special functions such as the exponential integral
$\text{ei}(u)=\int\frac{u'}{u}e^u\,dx$
and the error function*
$\text{erf}(u)=\int u'e^{u^2}\,dx$

* The usual error function, $\text{Erf}(x)=\int_0^xe^{-t^2}\,dt$ [Bate53], differs from our definition, which is denoted as Erfi in [Bate53], as follows: $\text{Erf}(x)=1/i\text{Erfi}(ix)$ . Also see the Appendix.

I wanted to know how to interpret the notation here (in the 'body' rather than the footnote, which I included to perhaps aid interpretation). What does it mean to take the derivative of u when it is a function argument? How do I take the integral wrt x of a seemingly-constant expression and not end up with x in the answer?

If more context (or the referenced Appendix) is needed I'll post more, but looking those over didn't help me understand.

R.D. Silverman · 2009-05-14, 14:24

Quote:

Originally Posted by CRGreathouse

From a paper I've been reading:

I wanted to know how to interpret the notation here (in the 'body' rather than the footnote, which I included to perhaps aid interpretation). What does it mean to take the derivative of u when it is a function argument? How do I take the integral wrt x of a seemingly-constant expression and not end up with x in the answer?

If more context (or the referenced Appendix) is needed I'll post more, but looking those over didn't help me understand.

The definitions that you quoted are simply wrong. You quote
definitions of functions of u, but the RHS integrals are with respect to
x, thus the RHS's are functions of TWO variables.

EI(u) = - INTEGRAL FROM -u to infinity of exp(-t)/t dt

CRGreathouse · 2009-05-14, 15:39

Quote:

Originally Posted by R.D. Silverman

The definitions that you quoted are simply wrong. You quote
definitions of functions of u, but the RHS integrals are with respect to
x, thus the RHS's are functions of TWO variables.

I just quoted it exactly as it appears in the paper. It's hard for me to distinguish mistakes, abuses of notation, and Things I Don't Know.

The function in the footnote is defined more conventionally.

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