If f is meromorphic on U with only a finite number of poles, then f=\frac{g}{h} where g and h are analytic on U.
We say f is meromorphic, then f is defined on U except at discrete set of points S which are poles. If z_0 is such a point, then there exist m in integers such that (z-z_0)^mf(z)...
I found some Matlab code that works. However, I am not sure how to alter it for my needs.
How can I make the code work for this:
$$N_{t+1} = \frac{(1+r)N_t}{1+rN_t}$$
What needs to be changed?
%%% MAKES A COBWEB PLOT FOR A LOGISTIC MAP
% compute trajectory
a=3.0; %...
Suppose f : \mathbb{C}\to \mathbb{C} is continuous everywhere, and is holomorphic at every point except possibly the points in the interval [2, 5] on the real axis. Prove that f must be holomorphic at every point of C.
How can I go from f being holomorphic every except that interval to...
g is continuous function, g:[-\pi,\pi]\to\mathbb{R}
Prove that the Fourier Transform is entire,
G(z)=\int_{-\pi}^{\pi}e^{zt}g(t)dt
So,
G'(z) = \int_{-\pi}^{\pi}te^{zt}g(t)dt=H(z).
Then I need to show that G(z) differentiable for each z_0\in\mathbb{C}.
I need to show...
f is analytic on an open set U, z_0\in U, and f'(z_0)\neq 0. Show that
\frac{2\pi i}{f'(z_0)}=\int_C\frac{1}{f(z)-f(z_0)}dz
where $C$ is some circle center at $z_0$.
S0 ,f(z)-f(z_0) = a_1(z-z_0)+a_2(z-z_0)^2+\cdots with a_1=f'(z_0)\neq 0. But why can f(z)-f(z_0) be expanded this way?
Without using Cauchy's Integral Formula or Residuals, I am trying to integrate
\int_{\gamma}\frac{dz}{z^2+1}
Around a circle of radius 2 centered at the origin oriented counterclockwise.
\frac{i}{2}\left[\int_0^{2\pi}\frac{1}{z+i}dz-\int_0^{2\pi}\frac{1}{z-i}dz\right]
\gamma(t)=2e^{it}...
For all z inside of C (C the unit circle oriented counterclockwise),
f(z) = \frac{1}{2\pi i}\int_C \frac{g(u)}{u-z} du
where g(u) = \bar{u} is a continuous function and f is analytic in C. Describe fin C in terms of a power series.
\displaystyle f(z) = \frac{1}{2\pi i}\int_C...
This is a plankton herbivore model.
The dimensionalized model is
\displaystyle
\frac{dP}{dt} = rP\left[(K-P)-\frac{BH}{C+P}\right], \quad \frac{dH}{dt} = DH\left[\frac{P}{C+P} - AH\right]
where r, K, A, B, C, and H are positive constants.
The dimensions of K, P, B, H, C have to be...
How do I find all $z$ that satisfies:
z = \exp\left(2+3i\right)
I know the modulus has to be e^2 and the argument has to be 3 but where do I go from there?
I am trying to show that there are 3 nonzero steady states of
\frac{du}{dt}=ru\left(1-\frac{u}{q}\right)-\left(1-\exp\left(-\frac{u^2}{\varepsilon}\right)\right)=0
I have tried using Mathematica and Mathematica couldn't solve it.
I tried some algebra and that wasn't going anywhere so I am at a...
f(z) = |z|
By the Cauchy-Riemann equations,
u_x = \frac{x}{\sqrt{x^2+y^2}}
v_y = -v_x = 0
u_y = \frac{y}{\sqrt{x^2+y^2}}
Since the C.R. equations don't work at (0,0), how can show f(z) is not holomorphic at (0,0)?
\sum_{n=2}^{\infty}z^n\log^2(n), \ \text{where} \ z\in\mathbb{C}
\sum_{n=2}^{\infty}z^n\log^2(n) = \sum_{n=0}^{\infty}z^{n+2}\log^2(n+2)
By the ratio test,
\lim_{n\to\infty}\left|\frac{z^{n+3}\log^2(n+3)}{z^{n+2}\log^2(n+2)}\right|
\lim_{n\to\infty}\left|z\left(\frac{\log(n+3)}{ \log...
r,q are constants. I need to factor this equation such that there is a double root.
-\frac{r}{q}u^3+ru^2-\left(\frac{r}{q}+1\right)u+r=0
Are there any tricks for this because this just a nasty equation.
I don't know if that is a wise approach but:
(au+b)(cu+d)^2 =...
Trying to remember how to use the definition of a complex limit.
\lim_{\Delta z\to 0}\frac{f(z+h)-f(z)}{\Delta z}
f(z) = |z| = \sqrt{x^2+y^2}
\Delta z = \Delta x + i\Delta y
\lim_{\Delta x\to 0}\frac{\sqrt{(x+\Delta x)^2+(y+\Delta y)^2}- \sqrt{x^2+y^2}}{\Delta x}
Is that correct? Or do I...