Laurent Series Expansion - Flashcard set - Complex Analysis

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Expand $f(z) = \ln(z+1)$ at $z_0=0$

$z - \frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} + \cdots$

Expand $f(z) = \frac{1}{z-1}$ at $z_0=2$

$1 + (z-2) + (z-2)^2 + (z-2)^3 + \cdots$

Expand $f(z) = e^z$ at $z_0=0$

$1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots$

Expand $f(z) = \sin(z)$ at $z_0=0$

$z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \cdots$

Expand $f(z) = \frac{1}{z^3}$ at $z_0=1$

$1 - 3(z-1) + 6(z-1)^2 - 10(z-1)^3 + \cdots$

Expand $f(z) = \frac{1}{z^2+z}$ at $z_0=-1$

$-\frac{1}{2} + \frac{z+1}{2} - \frac{(z+1)^2}{2} + \frac{(z+1)^3}{2} + \cdots$

Expand $f(z) = \frac{z}{(z-1)^2}$ at $z_0=2$

$1 + 3(z-2) + 6(z-2)^2 + 10(z-2)^3 + \cdots$

Expand $f(z) = \frac{1}{z^2+1}$ at $z_0=i$

$\frac{1}{2i} - \frac{z-i}{4} - \frac{(z-i)^2}{8i} - \frac{(z-i)^3}{16} + \cdots$

Expand $f(z) = \frac{z}{1-z}$ at $z_0=-1$

$-\frac{1}{2} - \frac{z+1}{4} - \frac{(z+1)^2}{8} - \frac{(z+1)^3}{16} + \cdots$

Expand $f(z) = (z-2)e^z$ at $z_0=0$

$(z-2) + (z-2)z + \frac{(z-2)z^2}{2!} + \frac{(z-2)z^3}{3!} + \cdots$

Expand $f(z) = e^{2z}$ at $z_0=0$

$1 + 2z + 2^2\frac{z^2}{2!} + 2^3\frac{z^3}{3!} + \cdots$

Expand $f(z) = \cos(z)$ at $z_0=0$

$1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \cdots$

Expand $f(z) = \log(z)$ at $z_0=1$

$(z-1) - \frac{(z-1)^2}{2} + \frac{(z-1)^3}{3} - \frac{(z-1)^4}{4} + \cdots$

Expand $f(z) = \frac{1}{(z-4)^2}$ at $z_0=5$

$1 - 2(z-5) + 3(z-5)^2 - 4(z-5)^3 + \cdots$

Expand $f(z) = \frac{1}{z-2}$ at $z_0=3$

$1 - (z-3) + (z-3)^2 - (z-3)^3 + \cdots$

Expand $f(z) = \cosh(z)$ at $z_0=0$

$1 + \frac{z^2}{2!} + \frac{z^4}{4!} + \frac{z^6}{6!} + \cdots$

Expand $f(z) = \frac{e^z}{z}$ at $z_0=1$

$e - e(z-1) + \frac{e}{2}(z-1)^2 - \frac{e}{6}(z-1)^3 + \cdots$

Expand $f(z) = \frac{z}{(z-2)^3}$ at $z_0=1$

$-1 + 3(z-1) - 6(z-1)^2 + 10(z-1)^3 - \cdots$

Expand $f(z) = (z+1)e^{-z}$ at $z_0=0$

$(z+1) - (z+1)z + \frac{(z+1)z^2}{2!} - \frac{(z+1)z^3}{3!} + \cdots$

Expand $f(z) = \frac{1}{(z-3)^2}$ at $z_0=2$

$-1 - 2(z-2) - 3(z-2)^2 - 4(z-2)^3 + \cdots$

Expand $f(z) = \frac{1}{z^2+2z+2}$ at $z_0=-1+i$

$\frac{1}{2i} - \frac{z+1-i}{2} - \frac{(z+1-i)^2}{4i} - \frac{(z+1-i)^3}{8} + \cdots$

Expand $f(z) = \sec(z)$ at $z_0=0$

$1 + \frac{z^2}{2!} + 5\frac{z^4}{4!} + 61\frac{z^6}{6!} + \cdots$

Expand $f(z) = e^{z^2}$ at $z_0=0$

$1 + z^2 + \frac{z^4}{2!} + \frac{z^6}{3!} + \cdots$

Expand $f(z) = \sinh(z)$ at $z_0=0$

$z + \frac{z^3}{3!} + \frac{z^5}{5!} + \frac{z^7}{7!} + \cdots$

Expand $f(z) = \frac{1}{(z+1)^3}$ at $z_0=0$

$1 - 3z + 6z^2 - 10z^3 + \cdots$

Expand $f(z) = \tan(z)$ at $z_0=0$

$z + \frac{z^3}{3} + 2\frac{z^5}{15} + 17\frac{z^7}{315} + \cdots$

Expand $f(z) = \frac{z^2}{z+1}$ at $z_0=-1$

$-1 + (z+1) - (z+1)^2 + (z+1)^3 - \cdots$

Expand $f(z) = \frac{1}{(z-1)(z-2)}$ at $z_0=3$

$-\frac{1}{2} + \frac{z-3}{2} - \frac{(z-3)^2}{2} + \frac{(z-3)^3}{2} - \cdots$

Expand $f(z) = \frac{1}{z(z-1)}$ at $z_0=2$

-\frac{1}{2} + \frac{z-2}{4} - \frac{(z-2)^2}{8} + \frac{(z-2)^3}{16} - \cdots

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