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Jan
31
comment Finding the value of this integral $ \int_{-\pi/4}^{\pi/4}{ (\cos{t} + \sqrt{1 + t^2} }\sin^3{t}\cos^3 {t})dt$?
Very nifty !...
Jan
31
comment Finding the value of this integral $ \int_{-\pi/4}^{\pi/4}{ (\cos{t} + \sqrt{1 + t^2} }\sin^3{t}\cos^3 {t})dt$?
You could split the integral into two terms. The first term should be easy to integrate. For the second, you may find $\frac{1}{2}\sin(2t)=\sin(t)\cos(t)$ useful.
Jan
30
comment why $\int{\cos({\pi}t)} dt = \frac{1}{\pi}\sin({\pi}t)$?
When something new is "added" we can assume the old rules will work, but often they don't work meaning there's more going on than we know. One way to think is to always remember that the rules we know at present are some special case of more general rules. So e.g. your statement $\int\cos(x)dx=\sin(x)+C$ is a special way of writing $\int\cos(1\cdot x)=\sin(x)+C$, but what if the $1$ were some other number $a$. What general rule is there to handle such cases? It's not as straight forward as the case when $a=1$.
Jan
27
comment Prove convergence of $\sum _{n=1}^{\infty }\sin(1/n)/n$
Since $1/n\to 0$ as $n\to\infty$, maybe you could consider $\sin(1/n)$ in a neighbourhood of $0$, e.g. $\sin(1/n)\to1/n$ as $n\to\infty$...
Jan
22
comment Finding the result of an infinite sum
That's no reason to methodically downvote every element of a question though.
Jan
19
comment Name for “3D quadrilateral” shape?
Thank you Andrew. I will have a think about your comment.
Jan
19
comment Name for “3D quadrilateral” shape?
Just found this: I think maybe polyhedron might cover it (in general). A polyhedron with 6 faces is called a hexahedron. mathworld.wolfram.com/Hexahedron.html
Jan
19
comment How to evaluate the integral $\int_{0}^{\infty}\frac{\cos {(ax)}-\cos{(b x)}}{x^2 }dx$?
Can you not use a keyhole contour?
Jan
19
comment How to evaluate the integral $\int_{0}^{\infty}\frac{\cos {(ax)}-\cos{(b x)}}{x^2 }dx$?
Another form is $$-\frac{1}{2}I=\int_0^\infty \frac{\sin\left(\frac{1}{2}(a+b)x\right)\sin\left(\frac{1}{2}(a-b)x\right)}{x^2}‌​{dx}.$$
Jan
19
comment Calculating $\arg(-1+\sqrt 3 \cdot i)$
To get a sense of what's going on, it can be useful to plot your complex number $-1+\sqrt{3}i$ in the complex plane. Your argument $\theta$ is then the counter-clockwise angle from the positive $x$-axis, which should give you some trigonometric intuition as to what you need to do given the quadrant within which the complex number lies, e.g. $-1+\sqrt{3}i$ is in the second quadrant, so you you know it must be $\pi/2+\phi$ where $\phi$ can be computed using trigonometry and the complex diagram. Or you could compute it using $\pi-\phi'$...
Jan
17
comment Finding the result of an infinite sum
Why the downvote?
Jan
12
comment Finding the cartesian equation of the plane containing two given lines
Find the normal to the two lines, and use $n\cdot (r-a)=0$ where $a $ is a point on the plane and $r=(x,y,z) $ is a general point.
Jan
12
comment Convergence of $\int_0^{\infty}\sin (p(t))dt$
I think $p (t)=a+bt $ may be a start for investigation. Expand using trig identities. Then maybe keep adding terms e.g. $c_i t^i $ and see what happens.
Jan
11
comment Not getting what it means $v = ai + bj + ck$ for some vector $v$
For example, suppose you wanted to prove what $(x,y,z)+(u,v,w)$ was equal to. Well: $$(x,y,z)+(u,v,w)=ix+jy+kz + iu+iv+iw=i(x+u)+j(y+v)+k(z+w)=(x+u,y+v,z+w).$$ Ok, we intuitively know what $(x,y,z)+(u,v,w)$ is (if you've learned that somewhere), but how would go about proving it? You need to use your definitions and we define $(x,y,z)=ix+jy+kz$. Similarly we need to define what $i,j,k$ are and also what it means to write $ix$ etc. That's maths - there's no room for ambiguity ! ;-)
Jan
10
comment I would like to find the value of $f(z)$
You should use the substitution $z=re^{i\theta}=r\cos\theta+ir\sin\theta\equiv g(r,\theta)+i h(r,\theta)$.
Jan
8
comment Prove that $a^{n+m}=a^{n}a^m$, for real numbers
Could it be $e^{n\log a}$... but then what of $a\leq 0$...
Jan
6
comment Basic understanding of quotients of “things”?
Ah yes of course ... will update.
Jan
6
comment Reference for multiplicative norms
Thanks for the info. I own a copy of Grossman and Katz (1972) - fascinating stuff !
Jan
6
comment Reference for multiplicative norms
I think that's called a sub-multiplicative norm; not 100% sure though.
Jan
6
comment Reference for multiplicative norms
@gerw It's some binary operation $ \circ: V^2 \to V $ on some vector space $ V $.