Recent content by shen07

Hi, I am trying to find the exact solution of the Continuity Equation. Any Idea how can i start solving it, i need it for some calculation in Image Processing. $$\pd{C}{t}+\pd{UC}{x}+\pd{VC}{y}=0$$ Where $U$ and $V$ is velocity in $X$ and $Y$ direction. The initial condition is as...

I just edited it

A fluid motion has velocity $$\underline{u}=\sin{(at)}\hat{\imath}+\hat{\jmath} \times r +\cos{(at)}\hat{k}$$ I need to know what is $$\hat{\jmath} \times r$$ to find Vorticity and other things.

consider the following LMM $$y_n-\frac{3}{2}y_{n-1}+\frac{1}{2}y_{n-2}=h(\frac{1}{2}f_n+\frac{1}{4}f_{n-1}-\frac{1}{4}f_{n-2})$$ which is applied to the initial value problem $$y'(t)=y(t),0\leq{t}\leq{1}\\ and\\ y(0)=1$$ How do i find an expression for $$y_n$$, if the starting values are...

Hi guys, well i have the problem below, $$\int_{\gamma(0;1)}\frac{1}{\exp(iz)-1}\mathrm{d}z$$ so it is holormorphic in D'(0,1) as it has a point not holomorphic at z=0. Taking a Laurent Series in the form $$f(z)=\sum_{n=-\infty}^{\infty}C_n(z-0)^n$$ But i wil get...

Ahh that's exactly what i was looking for as answer, Thanks a lot.

No, i don't agree with you. when You Paremetrise $$\theta$$ on $$\gamma(0;1)$$ you have a circle centre 0 and radius 1. - - - Updated - - - there is a z surplus in ur Numerator.. its $$ \frac{2}{i}\ \int_{\gamma} \frac{\ d z}{b\ z^{2} + 2\ a\ z + b}\$$

I tried this out but then which root lied inside $$\gamma(0;1)$$ and how do i evaluate the residue using this expression. Or should i simply do a Laurent Series to Obtain the coefficient of $$C_{-1}$$

Hi, I am supposed to use residue calculus to do the following integral $$\int_{0}^{2\pi}\frac{1}{a+b\cos( \theta) } \mathrm{d}\theta$$ for |b|<|a| i have paremetrise it on $$\gamma(0;1)$$ that is $$z=\exp(i\theta), 0\leq\theta\leq2\pi$$ and obtain the following...

Hi i am confused of the following question. Suppose we have a Metric Space (X,d), where d is the usual metric. Now are the following subsets complete, if so why?? 1.$$X=[0,1]$$ 2.$$X=[0,1)$$ 3.$$X=[0,\infty)$$ 4.$$(-\infty,0)$$

I really think this theorem will help you: Theorem: A bounded sequence of \mathbb{R} has a convergent sub sequence. If a sequence X is bounded,all its sub-sequences will be bounded. Now since every sequence has a monotone sub-sequence (i.e either decreasing or increasing), X will also have a...

What about the Residues of Multiple Poles: Multiple pole at z=a order m Type 1: $$f(z)=(z-a)^{m}g(z),g \in H(D(a;r)):$$ $$\text{Res}\{f(z);a\}=\frac{g^{m-1}(a)}{(m-1)!}$$ Type 2: $$\text{Res}\{f(z);a\}=\text{Coefficient }C_{-1}\text{ of }\frac{1}{z-a}$$ in the Laurent expansion of f(z) about...

Suppose i have like a function $$f(z)=\frac{(z-i)^2}{(z^3+1)}$$ then using Type 2 is much easier here than using Type 1, $$Res\{f(z);-1\}=\frac{h(-1)}{k^{'}(-1)},\text{ where }h(z)=(z-i)^{2}\text{ & }k(z)=z^{3}+1$$ Using type 1 would complicate things,Right??

Hi Zaid, its a notation: g is holomorphic(H) in the Disc center a,radius r - - - Updated - - - What you mean to say is that, both are the same?? bt in what case should we use each type?

Hi guys i wanted to clear out some confusion, Suppose $\text{z=a}$ is a simple pole, my Professor classify it as follows: Simple Pole at z=a Type 1: $$f(z)=(z-a)^{-1}g(z),g\in H(D(a;r)):$$ $$Res\{f(z);a\}=g(a)$$ Type 2: $$f(z)=\frac{h(z)}{k(z)},h(a)\neq 0,k(a)=0,k^{'}(a)\neq 0:$$...