a) Let $\,f\,$ be an analytic function in the punctured disk $\,\{z\;\;;\;\;0<|z-a|
My solution and doubt: If we develop $\,f\,$ is a Laurent series around $\,a\,$ we get $f(z)=\frac{a_{-k}}{(z-a)^k}+\frac{a_{-k+1}}{(z-a)^{k-1}}+\ldots +\frac{a_{-1}}{z-a}+a_0+a_1(z-a)+\ldots \Longrightarrow$ $\Longrightarrow f'(z)=-\frac{ka_{-k}}{(z-a)^{k+1}}-\ldots -\frac{a_{-1}}{(z-a)^2}+a_1+...$ and since $\,\displaystyle{\lim_{z\to a}f'(z)}\,$ exists finitely then it must be that $a_{-k}=a_{-k+1}=...=a_{-1}=0$ getting that the above series for $\,f\,$ is, in fact, a Taylor one and thus $\,f\,$ has a removable singularity at $\,a\,$ .
My doubt: is there any other "more obvious" or more elementary way to solve the above without having to resource to term-term differentiating that Laurent series?
b) Evaluate, using some complex contour, the integral $\int_0^\infty\frac{\log x}{(1+x)^3}\,dx$
First doubt: it is given in this exercise the hint(?) to use the function $\frac{\log^2z}{(1+z)^3}$Please do note the square in the logarithm! Now, is this some typo or perhaps it really helps to do it this way? After checking with WA, the original real integral equals $\,-1/2\,$ and, in fact, it is doable without need to use complex functions, and though the result is rather ugly it nevertheless is an elementary function (rational with logarithms, no hypergeometric or Li or stuff).
The real integral with the logarithm squared gives the beautiful result of $\,\pi^2/6\,$ but, again, I'm not sure whether "the hint" is a typo.
Second doubt: In either case (logarithm squared or not), what would be the best contour to choose? I though using one quarter of the circle $\,\{z\;\;;\;\;|z|=R>1\}\,$ minus one quarter of the circle $\,\{z\;\;;\;\;|z|=\epsilon\,\,,0<\epsilon<
$(i)\,$ to get the correct limits on the $\,x\,$-axis when passing to the limits $\,R\to\infty\,\,,\,\epsilon\to 0\,$
$(ii)\,$ To avoid the singularity $\,z=0\,$ of the logarithm (not to mention going around it and changing logarithmic branch and horrible things like this!).
Well, I'm pretty stuck here with the evaluations on the different segments of the path, besides being baffled by "the hint", and I definitely need some help here.
As before: these exercises are supposed to be for a first course in complex variable and, thus, I think they should be more or less "elementary", though this integral looks really evil.
For the time you've taken already to read this long post I already thank you, and any help, hint or ideas will be very much appreciated.