Compute $$\int_0^\infty\sin(x^2)\ dx,\ \int_0^\infty\cos(x^2)\ dx$$ using contour integral. I chose the contour integral over a sector cornered at $0,R$ and $Re^{i\pi/4}$. Let $f(z) = e^{iz^2}$ which is entire. Then, $$\int_0^R e^{ix^2}\ dx+\int_0^{\pi/4}e^{iR^2e^{2i\theta}}\ d\theta -\int_0^R ...

I am studing for exams and am stuck on this problem. Suppose $f$ is an entire function s.t. $f(z) =f(z+1)$ and $|f(z)| < e^{|z|}$. Show $f$ is constant. I've deduced so far that: a) $f$ is bounded on every horizontal strip b) for every bounded horizontal strip of length greater than 1 a ...

Let me try: Let $0<\epsilon<1$. Then Claim: If $f$ is entire with $f(z) =f(z+1)$ such that $$|f(z)|\leq e^{\epsilon|z|},$$ then $f$ is constant. $(\because)$ Since $f$ is $1$-periodc, $g(z) = f\left({\log z\over 2\pi i}\right)$ is a well-defined holomorphic function except for the origin regardle...

Suppose that $(X, S, \mu)$ is a measure space and that $f_n:X\to \mathbb R$ is a sequence of functions converging pointwise to $f$ in $L^\infty(\mu)$. It is given that $\mu(X)<\infty$. Then, I want to prove that $f_n\to f$ in $L^p(\mu)$ for any $p\in [1,\infty)$. Remark: This problem has an addi...