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Jun
8
comment The value of $\int^{\pi/2}_0 \frac{\log(1+x\sin^2\theta)}{\sin^2\theta}d\theta$
Sorry my mistake - updated.
Jun
8
revised The value of $\int^{\pi/2}_0 \frac{\log(1+x\sin^2\theta)}{\sin^2\theta}d\theta$
deleted 13 characters in body
Jun
8
answered The value of $\int^{\pi/2}_0 \frac{\log(1+x\sin^2\theta)}{\sin^2\theta}d\theta$
Jun
2
comment Why do we assume the complex plane is curvey at infinity?
@ Ollie Ford yes, I think that's what popped into my head, but mathematics needs to be rigorous, so this is just food for thought really !
Jun
2
comment Why do we assume the complex plane is curvey at infinity?
Just a thought: Give me any point $z$ in the upper half complex plane and I can tell you that the semi-circular contour of radius $|z|+1$ encloses it. Rectangular regions can be constructed similarly.
Jun
2
comment Extension of $|\cdot|_\infty$ on $\mathbb R$ to $\mathbb C$
I think this is related to Ostrowski's Theorem. Check out en.wikipedia.org/wiki/Ostrowski%27s_theorem "...any field, complete with respect to an archimedean absolute value, is (algebraically and topologically) isomorphic to either the real numbers or the complex numbers. This is sometimes also referred to as Ostrowski's theorem."
Jun
1
accepted Name for norm with property $\|x+y\|=\|x\|+\|y\|$.
Jun
1
comment Name for norm with property $\|x+y\|=\|x\|+\|y\|$.
Obviously people are liking your answer, but could you please clarify: Are you saying such a norm is called a "trivial norm"?
Jun
1
asked Name for norm with property $\|x+y\|=\|x\|+\|y\|$.
May
31
comment How do I calculate this integral:$\int_{0}^{1}\ln^2 \left| \sqrt x-\sqrt{1-x} \right|dx$?
I would split the domain to which $x$ belongs to describe positive and negative regions. That way you can dispose of the absolute value and simply evaluate a finite sum of integrals having no absolute values.
May
27
accepted Multiplying two inequalities
May
27
asked Multiplying two inequalities
May
27
revised How should I self-study calculus?
added 186 characters in body
May
27
answered How should I self-study calculus?
May
26
comment For which values of $\alpha$ does $ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^\alpha$ have a solution in integers?
Just some workings that may or may not lead to something useful: Take logarithms, $$\alpha\log\left(\frac{x+y}{3}+1\right)=\log(x^2+xy+y^2)$$ so that $$\alpha=\frac{\log(x^2+xy+y^2)}{\log\left(\frac{x+y}{3}+1\right)}.$$ Hence, you need $$x^2+xy+y^2=e^a$$ and $$\frac{x+y}{3}+1=e^b$$ such that $b\mid a$...
May
26
comment Rouché's Theorem for $p(z)=z^7-5z^3+12$
I may be wrong, but is your conclusion correct: $p(z)$ has $7$ roots in $D_2=\{z\in\mathbb{C}\mid |z|<2\}$ ? From your statement of Rouche's theorem you should conclude only that $p(z)$ has $0$ roots in $D_1$ since $f(z)=12$ has $0$ roots in $D_1$.
May
23
comment Is there an object in reality that is proven to be uncountable?
you could travel around the Earth countably infinite number of times, with certain unrealistic assumptions. That doesn't really answer your question but may be helpful to a layperson to understand you can pair off each circumnavigation with the natural numbers...
May
23
comment $\int _{ 0 }^{ 1 }{ \frac { { x }^{ t }-1 }{ \ln { x } } dx } $
What is $t$? An integer, rational, real, complex, ... Also, what have you tried so far?
May
22
accepted A name for the property $ \| x \star y \| = \| x \| \| y \| $.
May
19
comment Curious formula for minimum?
Thanks. I knew that must be true since they are clearly equal.