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Jun
28
comment Integral with only a list of values
I was a bit vague about the meaning of "under" because: suppose $a<b<c<d$ and suppose $f(a<x\leq b),f(c\leq x<d)\geq 0$ and $f(b\leq x\leq c)<0$. Then the the integral on $[a,d]$ does not define the area "under" (i.e. between the function and the x-axis), since the contribution of the integral on $[b,c]$ will be negative. Area is by definition always positive, so we need to be careful. Also, consider $\int_0^1 x^{−1/2}$. We have $\lim_{x→0}x^{−1/2}=\infty$, but the area under the curve is perfectly finite.
Jun
26
comment Solving $2^x - 3^x + 6^x =0$.
Thanks for your answer. I will take a look later when i get a spare bit of time.
Jun
26
comment Solving $2^x - 3^x + 6^x =0$.
Mathematica and WA are rather buggy in my experience, even some algebraic manipulations produce incorrect results.
Jun
26
comment Solving $2^x - 3^x + 6^x =0$.
@Did you originally stated $-x $ in the exponents... maybe that's it? On the move so can't check...
Jun
25
comment Find all complex numbers so that $Im(\frac{z+2}{2-i})=1$ and $Re(z^2+1)=1$ and for $z$ which is in the first quadrant find $\sqrt{z}$
I think you made a mistake in your 5th line down (see @mathlove's answer)
Jun
25
comment Looking for a bound on a function involving $\sinh$
Having played around with the function's Taylor expansion I think I can say that $\text{sech}(ct)$ is an upper bound... not proved it though... not sure if this helps with your question any either !
Jun
24
comment Need Suggestions for beginner who is in transition period from computational calculus to rigorous proofy Analysis
I'd recommend reading How to Think Like a Mathematician by Kevin Houston first. I wish I had done!
Jun
24
comment Riemann Zeta Function and Including Complex Numbers
See also this question which shows an analytic continuation of the Riemann zeta function to $\Re (s)>0$... math.stackexchange.com/questions/256992/…... an improvement on $\Re(s)>1$, but still not the whole story !
Jun
24
comment My brother asked me to explain a algebra problem. How should I explain it?
You're very welcome ! Welcome to SE.
Jun
21
comment Why $\sum\limits_{n=1}^\infty \frac{e^{inx}}{n}=-ln(1-e^{ix})$ in $D'$
Not quite sure about the question, but maybe this will help: Your sum is just $\sum_{n=1}^\infty\frac{(e^{ix})^n}{n}$, and $\sum_{n=1}^\infty \frac{z^n}{n}$ is a well known power series...
Jun
9
comment Why is $e$ so special?
Thinking aloud: the function $e^x$ has many interesting properties, e.g. if we consider the function $e^x$, then we find the derivate of $e^x$ is itself $e^x$. This is quite remarkable ! When $x=1$ we obtain $e^1=e$. $e^x$ is also strictly positive for all $x\in\mathbb{R}$. $e^x$ is also a transcendental function. Another remarkable formula involving $e^x$ is $e^{i\pi}+1=0$, where $i=\sqrt{-1}$.
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
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
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"?
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
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...