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Jan
11
revised Find appropriate substitution for indefinite integral.
edited body
Jan
8
revised How to calculate $ \int_0^1 \frac{(1+x)^{2r-1}}{1+x^2}dx $?
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Jan
8
revised How to calculate $ \int_0^1 \frac{(1+x)^{2r-1}}{1+x^2}dx $?
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Jan
7
answered How to calculate $ \int_0^1 \frac{(1+x)^{2r-1}}{1+x^2}dx $?
Jan
7
comment Is there a domain “larger” than (i.e., a supserset of) the complex number domain?
There are also the Quarternions ($z=a+ib+jc+kd\equiv(a,b,c,d)$), Octonions, and so on, each of which is a generalisation of the complex numbers. Each generalisation, however, loses some nice property of the "previous" set of numbers, e.g. commutatvity, and so on. I don't recall the exact details.
Jan
6
comment How to calculate $ \int_0^1 \frac{(1+x)^{2r-1}}{1+x^2}dx $?
Using the binomial theorem $$I=\sum_{k=0}^{2r-1}{2r-1\choose k}\int_0^1 \frac{x^k}{1+x^2}dx.$$ Taking $k$ even and odd separately gives "nice" values, although I don't have time to work out their closed forms right now. What have you tried so far?
Dec
22
comment Nice approximations of sums by integrals.
Have you tried using the Euler-Maclaurin summation formula? A Google search will give further details (en.m.wikipedia.org/wiki/Euler–Maclaurin_formula)
Dec
20
accepted Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?
Dec
20
revised Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?
edited title
Dec
20
revised Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?
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Dec
20
asked Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?
Dec
19
awarded  Constituent
Dec
16
comment Is the derivative of $\{x\}$ on $(0,1)$ always equal to $1$?
Yes, sorry - question updated. Thanks for spotting this.
Dec
8
awarded  Caucus
Dec
7
awarded  Nice Answer
Nov
26
awarded  Notable Question
Nov
20
comment Finding the roots of $x^2+(3+5i)x+(7+11i)=0$
Re polar exponential form - see my answer below for the answer.
Nov
20
revised Finding the roots of $x^2+(3+5i)x+(7+11i)=0$
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Nov
20
revised Finding the roots of $x^2+(3+5i)x+(7+11i)=0$
added 542 characters in body
Nov
20
answered Finding the roots of $x^2+(3+5i)x+(7+11i)=0$