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You want to show that $$\int_{-\pi}^\pi|\log|1+e^{-it}||dt$$ is finite. Now $$|1+e^{-it}|=|e^{it/2}+e^{-it/2}|=2\cos(t/2)$$ so your integral is $$\int_{-\pi}^\pi|\log|2\cos(t/2)||dt =2\int_0^\pi|\log| …

It follows from Lebesgue's characterization of Riemann integrable functions as bounded functions continuous outside a set of Lebesgue measure zero.

This is discussed briefly as a generalization of Shapiro's cyclic sum inequality by J. Michael Steele in his book The Cauchy-Schwarz Master Class. He remarks that (1.) holds for $n\ge25$ and refers to …

It's certainly the case that $\mathbb{R}^2\setminus J$ is path connected. So any two points in $D\setminus J$ are joined by a path in $\mathbb{R}^2$ missing $J$. If this path isn't in $D$ it hits the …

See Conway's base 13 function.

By Taylor's theorem $$\phi(t+h)=\phi(t)+h\phi'(t)+h^2\phi''(t+u(h,t)h)/2$$ where $0\le u(h,t)\le1$. So $$\phi_h(t)=\phi'(t)+h\phi''(t+u(h,t)h)/2.$$ As $\phi''$ is in $C^\infty_c$ it's pretty clear tha …

It's a divergent infinite product. You might as well ask for the sum of $$\sum_{n=1}^\infty\frac{x}{n\pi}.$$ You can "cure" the divergence by multipliying each term by a suitable factor, so for instan …

To give a solution from scratch, start with the identity $$\sum_{s=-\infty}^\infty\frac1{(x+s)^2}=\frac{\pi^2}{\sin^2\pi x}.$$ We want to prove that this extends to an identity $$\sum_{s=-\infty}^\inf …

It is a standard result that each closed subset of $\mathbb{R}^n$ is a zero set of some smooth function on $\mathbb{R}^n$. One proves this using smooth bump functions and partitions of unity.

As other correspondents have pointed out, this is essentially a theta function. You ask if you can write it in any other way. You can replace the infinite series by an infinite product :-) One gets $$ …

There are proofs in textbooks such as Lang's Real and Functional Analysis and Conlon's Differentiable Manifolds but they do use some Banach space theory (not an awful lot). I have also seen in one wel …