(PROBLEM SOLVED)
I am trying to think of a complex function that is nowhere differentiable except at the origin and on the circle of radius 1, centered at the origin. I have tried using the Cauchy-Riemann equations (where f(x+iy)=u(x,y)+iv(x,y))
\frac{\partial u}{\partial x}=\frac{\partial...
Ah!! Thank you very much for that insight, it hit me as soon as I read what you had to say. I haven't used that trick in awhile so I think I temporarily forgot about it.
I have just learned the residue theorem and am attempting to apply it to this intergral.
\int_{0}^{\infty}\frac{dx}{x^3+a^3}=\frac{2\pi}{3\sqrt{3}a^2}
where a is real and greater than 0. I want to take a ray going out at \theta=0 and another at \theta=\frac{2\pi}{3} and connect them with an...
Well think of the alternating series, something like
\sum_{k=1}^{\infty} (-1)^n a_k
with a_k always positive. This will jump back and forth from negative to positive on the number line as it progresses so most of the time a comparison test with these wont work (you would need to find...
Yes, if after large k the series fits the necessary condition of being larger or smaller. Of course this cannot be used for some obvious series which wont work, like alternating, which have their own tests.
Well the comparison test says that if you have 0\leq a_k \leq b_k, then if
\sum_{k=1}^{\infty}b_k<\infty \quad \Rightarrow \quad \sum_{k=1}^{\infty}a_k <\infty
and if
\sum_{k=1}^{\infty}a_k=\infty \quad \Rightarrow \quad \sum_{k=1}^{\infty}b_k=\infty
So you need to find either a...
Just use the Taylor series to expand it about whatever point a;
\sum_{n=0}^{n=\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n
Looks like for \frac{x}{(1-x)^2} this it will be something like
x+2x^2+3x^3+4x^4+\ldots+
if you center it on 0.
Yea that integral is messy. I tried to get a numerical integration of the entire double integral you wanted and it's complex; I get
\int_0^1\int_{x^2}^1xe^{xy} dy dx = 0.376377+ (3.9255\times 10^{-16})\imath
Well an integral like that is not going to be integrable in terms of elementary functions. I know that something like...
\int e^{-x^2}dx
is an integral that is a gaussian distribution which you can find in a table but that's the closest thing I can think of to the integral you mentioned.
An interesting inequality--question on the proof...
I am working on the following proof and have gotten about half way;
If a_1, a_2, \ldots , a_n are positive real numbers then
\sqrt[n]{a_1 \cdots a_n} \leq \frac{a_1+a_2+\ldots +a_n}{n}
By induction, I started by showing it for n=2...
I'm working out of Abbott's Understanding Analysis and I'm trying to show the following,
For an arbitrary function g :\mathbb{R}\longrightarrow \mathbb{R} it is always true that g(A\bigcap B) \subseteq g(A) \bigcap g(B) for all sets A, B \subseteq \mathbb{R}.
I'm confused on how to get...