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    If f is meromorphic on U with only a finite number of poles, then

    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)...
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    Cobb web Matlab

    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; %...
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    Trivial power series question

    How do I find the power series for z^7? I can't remember.
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    Holomorphic on C

    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...
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    Fourier Transform

    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...
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    Cauchy Integral Formula application

    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?
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    Complex integration

    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}...
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    Cauchy Integral Formula

    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...
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    Nondimensional DE

    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...
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    Complex powers

    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?
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    Hysteresis and steady states

    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...
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    Convergence help

    \sum_{n=0}^{\infty}\left(\frac{z}{z+1}\right)^n z in C. This only converges for z>\frac{-1}{2}, correct? Thanks.
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    Modulus of z

    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)?
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    Find the power series

    z is a complex number. f(z) = \frac{4 + 3z}{(z + 1)(z + 2)^2} \frac{1}{1 + z} = \frac{1}{1 - (-z)} = 1 + (-z) + (-z)^2 + \cdots = \sum_{n = 0}^{\infty}(-z)^n \frac{1}{(z + 2)^2} = -\frac{d}{dz} \ \frac{1}{1 - (-z - 1)} = -\frac{d}{dz}\sum_{n = 0}^{\infty}(-z - 1)^n = \sum_{n =...
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    Power series

    z\in\mathbb{Z} \frac{1}{1-(-z)}=\sum_{n=0}^{\infty}(-z)^n \frac{1}{(z+2)^2}=\frac{d}{dz} \frac{-1}{1-(1-z)} = \frac{d}{dz} (1 + (1-z) + (1-z)^2+\cdots = 0 -1 -2(1-z)-3(1-z)^2 - \cdots = \sum_{n=0}^{\infty} ??? Not to sure about the second one.
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    Ratio Test

    \displaystyle\sum_{n = 1}^{\infty}\frac{(n!)^3}{(3n)!}z^n , \ z\in\mathbb{C} By the ratio test, \displaystyle L = \lim_{n\to\infty}\left|\frac{[(n + 1)!]^3 z^{n + 1} (3n)!}{[3(n + 1)]! (n!)^3 z^n}\right| = \lim_{n\to\infty}\left|\frac{z (n + 1)^2}{3}\right| = \infty. Therefore, the series...
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    Radius of convergence

    \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...
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    Double root

    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 =...
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    Complex Limit

    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...
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    Complex Analysis

    I am trying to decipher what this means: F(z) = \overline{f(\bar{z})} Thanks for the help.