Vtyjkyuk

I am studying the book Gamma by Julian Havil and there are three equations below stated without proofs:

beginalign
operatorname Si(x) & = sum_k = 1^infty (-1)^k - 1 fracx^2k - 1(2k-1)(2k-1)!, \[10pt]
operatornameCi(x) & = -gamma - ln x -sum_k=1^infty frac(-x^2)^k(2k)(2k)!, \[10pt]
operatornameLi(x) & = gamma + ln ln x + sum_k=1^infty fracln^r xkk!.
endalign

I couldn't find any proof for the three equations above.

A clear proof for each either in MSE or a book containing them would be much appreciated.

The only book I know of that directly derives these formulae is Theorie Des Integrallogarithmus Und Verwandter Transzendenten by Niels Nielsen. This is a German language book published in 1906. It is freely available online if I remember correctly.

But also note that one can alternatively define

$$operatornameSi(x) = int_0^x fracsintt , textdt$$

so that term-by-term integration of the Taylor series for $sin$ yields

$$operatornameSi(x) = sum_n=1^infty frac(-1)^n-1 x^2n-1(2n-1)(2n-1)!$$

The trickier part is to show that

$$operatornameCi(x) = -int_x^infty fraccos tt , textdt = gamma + log x + int_0^x fraccos t-1t , textdt$$

so that one can integrate term-by-term to arrive at
$$operatornameCi(x) = gamma + log x + sum_n=1^infty frac(-1)^n-1 x^2n2n(2n)!$$

But this is all done in Nielsen using $operatornameLi$.

Edit: I realize this book is not great for self-study, but it is truly a gem and if you are interested in this sort of mathematics it is a great resource.