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I wanted to know how to interpret the notation. 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? Generally, how are these functions (typically defined with definite integrals) defined with indefinite integrals? (Or is this just a trick of notation?)This class includes, in addition to the elementary functions, a number of well-known special functions such as the exponential integral

[tex]\text{ei}(u)=\int\frac{u'}{u}e^u\,dx[/tex]

and the error function*

[tex]\text{erf}(u)=\int u'e^{u^2}\,dx[/tex]

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

I'd be happy to post more of the paper if it would help. I'm actually recopying it at the moment, so it wouldn't be too hard.