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Banach space is the standard setting for the French Analysis courses of the second half of XX century, for example H. Cartan, Calcul differentiel. Formes differentielles, Herman, Paris, 1967. It does …

I do not know much about categories, but I interpret the question as a general question: how to find out whether some mathematical fact is known/true. Here I strongly agree with Rodrigo: ask an expert …

Good reference in English is Hubbard, Teichmuller theory, Matrix editions, Ithaca, NY, 2006, MR2245223. There is a very good reference in French, H. P. de Saint Gervais, Uniformisation des surfaces de …

All works of Chebyshev are available in French (and Russian, of course). There is a recent book MR2106657 M. Bronstein, Symbolic integration. I., which must contain this (sorry I could not check; do …

People have difficulties with Boutroux asymptotics not only because his French:-) And the matter is too complicated to be explained in the space allowed for answers in this site. You may look at the …

There are such proofs. See, for example Goluzin, Geometric theory of functions (Appendix). He uses the following fact. Let $h$ be an analytic diffeomorphism of the circle onto itself. Then there is a …

You wrote a series. If you mean that it converges for all $x\in R^n$ than this is an entire function that is the series also converges for $x\in C^n$. An entire algebraic function is a polynomial, ind …

I do not know of any survey books of this sort (perhaps it is impossible to write because "engineering" is too large and diverse) but I have two outstanding mathematical problems arising from engineer …

An example which always puzzled me is J. Milnor's paper entitled Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 1964, 542. The whole paper occupies about ha …

The place to start is: Hormander, Analysis of Linear partial differential operators, vols. I-II (if your coefficients are constant), and further volumes for non-constant coefficients.

Unlike Mathscinet, Zentralblatt usually reviews translations. I checked Zentralblatt, and found no translation of this work. So probably English translation does not exist. Actually very few works of …

First of all, the problem can have no solution. Consider intervals $I_n=(0,n)$. No matter how you color them with $k$ colors there will be infinite families with same color and non-empty intersection. …

Edit. Let us first prove the inequality: $$\sup_{z,w}|f(z)-f(w)|\geq\sup_z|f(z)-f(-z)|=\sup|g(z)|\geq|g'(0)|=2|f'(0)|,$$ where the Schwarz Lemma was applied to $g(z)=f(z)-f(-z)$. Equality in Schwarz …

There are many modern books, for example, MR1317343 Balser, Werner From divergent power series to analytic functions. Theory and application of multisummable power series. Lecture Notes in Mathemati …

In two dimensions, you have a powerful tool, the Riemann mapping. If you have a compact set in the plane whose complement is connected, knowing explicitly the map of the complement onto the exterior o …