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Dec
16
accepted Calculate $I_m = \int_{-\infty}^\infty \frac{dx}{1+x+x^2+\cdots+x^{2m}}$ using complex variables
Dec
15
accepted What conditions are necessary on $a,b,c,d$ so that the Mobius transformation $w=\frac{az-b}{cz-d}$ has only one fixed point?
Dec
14
accepted Let $z_1$, $z_2$ and $z_3$ be complex vertices of an equilateral triangle. Show $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$.
Dec
13
accepted How can I show that $u=e^{\sigma\sqrt{\Delta t}}$ in the binomial option pricing model
Dec
12
accepted Describe the Riemann surface for $w=z^2-1$.
Dec
9
accepted Without using beta function, for what $p$ does $\int_0^\infty \frac{\log(x)}{1+x^p} dx$ converge?
Dec
9
accepted For what values $p$ does $\int_0^\infty \frac{\log(x)}{1+x^p} dx$ converge.
Dec
9
accepted Show the level curves of $\log|f(z)|$ are orthogonal to those of $\operatorname{arg}(f(z))$.
Dec
6
accepted Without using the beta function, find values $q,r$ such that the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge?
Dec
4
accepted Prove $f'(x) \geq x f(x)$ $\forall x \in \mathbb{R}$ $\implies$ $\exists k$ s.t. $ke^{x} \leq f(x)$ $\forall x \in \mathbb{R}$.
Dec
2
accepted $\{f_n\} \in \mathcal{R}[a,b]$, converges pointwise to $f\in \mathcal{R}[a,b]$. Prove $\lim_{n\to\infty}\int_a^b f_n \, dx = \int_a^b f \, dx$.
Dec
2
accepted If $f: \mathbb{R} \to \mathbb{R}$ is continuous then $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$
Dec
2
accepted Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$.
Dec
2
accepted Sequence of $f_n \in R[0, 1]$ that converges pointwise to $f \in R[0, 1]$ such that $\lim_{n \to \infty} \int_0^1 f_n dx \neq \int_0^1 f dx$.
Nov
26
accepted Find sequence of differentiable functions $f_n$ on $\mathbb{R}$ that converge uniformly, but $f'_n$ converges only pointwise
Nov
24
accepted What Mobius transformation maps the circles $|z-\frac{1}{4}| = \frac{1}{4}$ and $|z|=1$ onto two concentric circles centered at $w=0$?
Nov
24
accepted Find the linear fractional transformation that maps the circles |z-1/4| = 1/4 and |z|=1 onto two concentric circles centered at w=0?
Nov
22
accepted Statement of maximum modulus principle and question
Nov
20
accepted Which mobius transformations map $|z-1|=1$ and $|z+1|=1$ onto the lines $Re(w)=1$ and $Re(w)=-1$, respectively, and the single point $z=2$ onto $w=1$?
Nov
20
accepted If $\alpha$ is of bounded variation on $[a,b]$, then it is continuous almost everywhere on $[a,b]$