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Apr
13
comment Integral of a square compared to the square of an integral
This can also be proven using Parsevals theorem.
Apr
13
comment Integral of a square compared to the square of an integral
What you are trying to prove is false. The $L^2$ norm does not equal the $L^1$ norm in general for constant functions. Notice that $\int C^2 =(b-a)C^2$ whereas $(\int C )^2 = (b-a)^2 C^2$.
Apr
13
comment Heuristic explanation for oscillatory behavior of first $n$ primes' multiples
What are you plotting? This question is not clear.
Apr
13
comment evaluate $\int \frac{\tan x}{x^2+1}\:dx$
I am going to guess that if this is coming from a homework, the intended question was $\int \text{arctan}(x)/(x^2+1)dx$ which has a nice form.
Apr
11
revised Asymptotics for square-free numbers in an arithmetic progression
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Apr
10
revised Asymptotics for square-free numbers in an arithmetic progression
edited title
Apr
10
answered Asymptotics for square-free numbers in an arithmetic progression
Mar
31
answered Are there infinitely many Thâbit ibn Kurrah cousin primes?
Mar
31
awarded  Enlightened
Mar
31
awarded  Nice Answer
Mar
30
awarded  Necromancer
Mar
30
revised Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$
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Mar
30
revised Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$
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Mar
30
revised Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$
added 4 characters in body
Mar
30
revised Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$
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Mar
30
revised Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$
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Mar
30
comment Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$
Lastly, if you are skeptical, I have added several exact computations, and you may verify numerically that these are correct.
Mar
30
comment Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$
@columbus8myhw: Yes, the rational number $c/d$ should be in lowest terms. I have added that. It cannot equal $1$, as the proof would breakdown in that case. I added the slightly stronger condition that in lowest terms, at least one of $c,d$ must be even. This is because I did not want to deal with the partial fraction expansion when there is a double root at $x=0$. It should not take much more work extend this result to all rational numbers, but the form may be slightly different when both $c,d$ are odd.
Mar
30
revised Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$
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Mar
30
comment Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$
@columbus8myhw: I had never looked at this question before.