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Jan
21
comment Logarithms of Negative Numbers
$\log z=\log|z|+i\text {Arg} z $ if you take the principal branch. Exponentiating both sides gives $ z=|z|e^{i\text{Arg}( z)} $ as you might hope to expect.
Jan
20
comment Computing the integral of $\int \frac{25x^2}{(x+3)(x-2)^2}\,dx$
what is "impartial differentiation" ?
Jan
20
comment Proof of $\cos(y)$ and $\sin(y)$ using $e^{iy}$
A simple case of substitution followed by simplification will achieve what you want.
Jan
20
comment Solve the integral $\int \frac{x^3+1}{x^2+7x+12}\, dx$
What have you tried so far? Any ideas what might work?
Jan
20
comment Does $\int_0^\infty \sin(x^{2/3}) dx$ converges?
Maybe $$I=\Im\int_0^\infty e^{ix^{2/3}}dx,$$ and then use the error function. Looks divergent.
Jan
20
comment Equation $e^{\frac{1}{x}} - x =0$
Could we use Lagrange inversion theorem here? en.wikipedia.org/wiki/… c.f. also @ PM's comment and @ Lukas' answer.
Jan
19
comment Is there a name for functions “opposite in nature” to orthogonal functions?
Thanks David. Are you using the property that the square of a function is always positive here? What if $g_n^2$ is not always positive, e.g. $g_n(x)=\sqrt{h(x)}$ for some $ h $ whose image is $[-a, a] $ ?
Jan
17
comment Examples of orthogonal/orthonormal functions which are not finite degree polynomials?
Ah I see. Is this related to the so-called Schmidt process? (related to Gram-Schmidt by any chance...)
Jan
15
comment How to prove that $\left(\sqrt{3}\sec{\frac{\pi}{5}}+\tan{\frac{\pi}{30}}\right)\tan{\frac{2\pi}{15}}=1$
In general, squaring and applying known trigonometric identities can sometimes be of help. Might not be of help here though.
Jan
13
comment Memory efficient algorithm to find network diameter
So the graph has an embedding in the plane? Or just edge lengths?
Jan
13
comment Memory efficient algorithm to find network diameter
Why would it fill memory? Are you using an object oriented paradigm to represent your graph in memory? BFS of this graph representation has little overhead.
Jan
13
comment How do I find the equivalence of the expression $e^{n\log(n)-(n+e)\log(n + e)}$?
Do you mean you want to find an asymptotic formula for the expression?
Jan
13
comment A fan, a horn, and a snowflake - unusual math terms
The `Golden ratio' $\phi$.
Jan
13
comment Integer root of a quadratic
Do you mean over all $a\in\mathbb{Z}$ or just some specific $a\in\mathbb{Z}$ for which $n^2-an+6a=0$ has integer solution(s) $n$ ? If it's a specific $a$ then don't you just need to use the quadratic formula, assuming your integer $a$ gives integer solutions for $n$?
Jan
12
comment Gradient of product of sums
Maybe the product rule? $(u\cdot v)'=u\cdot v'+v\cdot u'$ ?
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
16
comment Is the derivative of $\{x\}$ on $(0,1)$ always equal to $1$?
Yes, sorry - question updated. Thanks for spotting this.
Dec
3
comment Summation of logarithms
You have $$S=\sum_{k=1}^n\log(a-x_k)=\log\prod_{k=1}^n(a-x_k).$$ So $$\prod_{k=1}^n(a-x_k)-e^S=0.$$ So you need to be able to solve equations of the form: $$\prod_{k=1}^n(a-x_k)-c=0,$$ where $c$ is some constant.