# flapjackery

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# 90 Posts

 13h asked If $f: \mathbb{R} \to \mathbb{R}$ is continuous then $\{ x \in \mathbb{R} \mid f(x) > 0\}$ is an open subset of $\mathbb{R}$ 1d asked Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$. 1d asked Prove $f'(x) \geq x f(x)$ $\forall x \in \mathbb{R}$ $\implies$ $\exists k$ s.t. $ke^{x} \leq f(x)$ $\forall x \in \mathbb{R}$. 2d asked Find sequence of differentiable functions $f_n$ on $\mathbb{R}$ that converge uniformly, but $f'_n$ converges only pointwise 2d asked Sequence of $f_n \in R[0, 1]$ that converges pointwise to $f \in R[0, 1]$ such that $\lim_{n \to \infty} \int_0^1 f_n dx \neq \int_0^1 f dx$. Nov22 asked Statement of maximum modulus principle and question Nov20 asked 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? Nov20 asked For what values $q,r$ does the improper integral $\int_0^1 x^q (1-x^2)^r dx$ converge? Nov20 asked If $\alpha$ is of bounded variation on $[a,b]$, then it is continuous almost everywhere on $[a,b]$ Nov19 asked For what values $p,q$ does the improper integral $\int_0^1 x^p (1-x^2)^q dx$ converge? Nov18 asked Show $\int_{C_N} \frac{\pi \sec(\pi z)}{z^3} dz \to 0$ for given contour. Nov17 asked Find the residues of $F(z) = \frac{\pi \sec \pi z}{z^3}$ at $z = n+ \frac{1}{2}$, $n = 0, \pm 1, \pm 2, \ldots$ Nov17 asked Find all possible linear fractional transformations that map circles $|z-1|=1$ and $|z+1|=1$ onto lines $Re(w)=1$ and $Re(w)=-1$ and $z=2$ to $w=1$. Nov17 asked What Mobius transformation maps the circles $|z-\frac{1}{4}| = \frac{1}{4}$ and $|z|=1$ onto two concentric circles centered at $w=0$? Nov16 asked 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 $w=1$? Nov16 asked If $\alpha,\beta \in BV[a,b]$, prove that $|a| \in BV[a,b]$ and $\min(\alpha,\beta),\max(\alpha,\beta) \in BV[a,b]$. Nov15 asked Compute $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3} = \frac{\pi^3}{32}$ using residue theory. Nov13 asked Using complex analysis, calculate $I_m = \int_{-\infty}^\infty \frac{dx}{1+x+x^2+\cdots+x^{2m}}$ for $I_2$ and $I_3$ Nov11 asked $f:[0,1] \to \mathbb{R}$ with $f(0) = 0 = f(1)$ and $|f''|\leq M$. Show $|f'(1/2)| \leq \frac{M}{4}$. Nov11 asked Find a polynomial which approximates $f(x) = \sqrt{x}$ in the interval $(4,5)$ within $10^{-8}$