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recall the MacLaurin series + \[\frac{1}{1 - z} = \sum^{\infty}_{n=0} (-1)^nz^n \] + which converges for $|z| < 1$. Applying the change of variables $z = \frac{z}{3}$, + we get + \[\frac{1}{1 - z/3} = \sum^{\infty}_{n=0} (-1)^n (\frac{z}{3})^n \] + which converges for $|\frac{z}{3}| < 1 \implies |z| < 3$. + + \bigskip + + Applying all of this, we get + \[\begin{aligned} + f(z) + &= \frac{1}{3z^2} \cdot \frac{1}{1 - \frac{z}{3}} \\ + &= \frac{1}{3z^2} \cdot \sum^{\infty}_{n=0} (-1)^n (\frac{z}{3})^n \\ + &= \frac{1}{3z^2} \cdot \sum^{\infty}_{n=0} (\frac{-1}{3})^n z^n \\ + &= \frac{1}{3} \cdot \sum^{\infty}_{n=0} (\frac{-1}{3})^n z^{n-2} + & (0 < |z| < 3)\\ + \end{aligned}\] + + \paragraph{Solution about $z = 1$} + \end{enumerate} + + \item[4.] + \begin{enumerate} + \item Find the MacLaurin series representation of $\cos{z} = \frac{e^{iz} - e^{-iz}}{2}$ + + \bigskip + + \[\cos{z} = \sum^{\infty}_{n=0} \frac{(-1)^n (z^{2n})}{(2^n)!} \] + + \begin{proof} + First, calculate and evaluate the first few derivatives of $\cos{z}$ at $z = 0$. + + \[\begin{aligned} + f^{(0)}(0) &= \cos{0} = 1 \\ + f^{(1)}(0) &= -\sin{0} = 0 \\ + f^{(2)}(0) &= -\cos{0} = -1 \\ + f^{(3)}(0) &= \sin{0} = 0 \\ + f^{(4)}(0) &= \cos{0} = 1 \\ + \end{aligned}\] + + As we can see, the derivatives of $\cos{z}$ follow a predictable pattern. Continuing, using the general formula for a MacLaurin series + \[f(z) = \sum a_n z^n\] + implies that + \[\cos{z} = \sum^{\infty}_{\text{$n$ odd}} 0 + + \sum^{\infty}_{\text{$n$ even}} \frac{(-1)^{\frac{n}{2}}}{n!} \cdot (z^n) \] + + Furthermore, if $n$ is even, then there is some integer such that $n$ is divisible by $2$, i.e. $n = 2n$. Using this change of variables + + \[\cos{z} = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} \cdot (z^{2n}) \] + + as we set out to prove. + \end{proof} + + \item Using the MacLaurin series representation for the function $\cos{z}$, find the MacLaurin series representation for $\sin{z}$ + \end{enumerate} + + \item[6.] Find the first three non-zero terms in the MacLaurin expansion of + \[f(z) = \int_0^{z} e^{s^2} ds\] + + \paragraph{Solution} + First, recall the MacLaurin Series + \[e^z = \sum^{\infty}_{n=0} \frac{z^n}{n!} \] + + Thus, letting $z = s^2$, we get + \[e^{s^2} = \sum^{\infty}_{n=0} \frac{{s^2}^n}{n!} \] + which, like the original series, converges for all $|z| < \infty$. + + Now, + \[\begin{aligned} + \int_0^z e^{s^2} ds + &= \int_0^z \sum^{\infty}_{n=0} \frac{{s^2}^n}{n!} ds \\ + &= \sum^{\infty}_{n=0} \frac{ + \int_0^z {s^2}^n ds }{n!} & \text{Integrate term by term} \\ + &= \sum^{\infty}_{n=0} \frac{s^{2n + 1}}{(2n + 1) \cdot n!} \\ + \end{aligned}\] + + Therefore, the first three non-zero terms are + \[ + \frac{s^{2(0) + 1}}{(2\cdot0 + 1) \cdot 0!} + + \frac{s^{2(0) + 1}}{(2\cdot1 + 1) \cdot 1!} + + \frac{s^{2(0) + 1}}{(2\cdot2 + 1) \cdot 2!} + \] + + \item[8.] Find residue at $z = 0$ of $\frac{1}{z^2 + z^3}$. + + First, recall the MacLaurin series + \[ \frac{1}{1+z} = \sum^{\infty}_{n=0} (-1)^n z^n \] + which converges for $|z| <1$. Thus, + \[\begin{aligned} + \frac{1}{z^2} \cdot \frac{1}{1+z} + &= \sum^{\infty}_{n=0} (-1)^n z^{n-2} & (|z| < 1) \\ + &= (-1)^0 z^{-2} + (-1)^1 z^{-1} + (-1)^2 z^0 + ... \\ + &= z^{-2} - \mathbf{z^{-1}} + ... \\ + \end{aligned}\] + + Therefore, $Res_{z=0} f(z) = 1$. + + \item[9.] + \begin{enumerate} + \item Use Cauchy's Residue Theorem to evaluate the integral around the circle + $|z| = 3$ with a positive orientation fo $z^3 \cdot \exp{\frac{1}{z^2}}$ + + \paragraph{Solution} To begin, we have an isolated singularity at $z = 0$. Then, recall the MacLaurin Series + \[e^z = \sum^{\infty}_{n=0} \frac{z^n}{n!} \] + + Using the change of variables $z = \frac{1}{z^2}$, we get + \[\begin{aligned} + z^3 \cdot \exp{\frac{1}{z^2}} + &= z^3 \sum^{\infty}_{n=0} \frac{(\frac{1}{z^2})^n + }{n!} \\ + &= z^3 \sum^{\infty}_{n=0} \frac{z^{-2n}}{n!} \\ + &= \sum^{\infty}_{n=0} \frac{z^{-2n + 3}}{n!} \\ + \end{aligned}\] + + Expanding this series, we get + \[\frac{z^3}{0!} + \frac{z^{-2 + 3}}{1!} + \mathbf{\frac{z^{-4+3}}{2!}} + ... + \] + + Therefore our residue is $2! = 2$. + + \item ... + \end{enumerate} + + \item[10.] Write the principal part of the following functions at their singular point and determine whether that point is a pole, a removable singular point ,or an essential singular point: + + \begin{enumerate} + \item $z \exp{\frac{1}{z^3}} $ + + \paragraph{Solution} First, notice that we have a singular point at $z = 0$. Then, recall the MacLaurin Series + \[e^z = \sum^{\infty}_{n=0} \frac{z^n}{n!} \] + + Using the change of variables $z = \frac{1}{z^3}$, we get + \[\begin{aligned} + z \exp{\frac{1}{z^3}} + &= z \cdot \sum^{\infty}_{n=0} \frac{( \frac{1}{z^3} )^n}{n!} \\ + &= z \cdot \sum^{\infty}_{n=0} \frac{z^{-3n}}{n!} \\ + &= \sum^{\infty}_{n=0} \frac{z^{-3n + 1}}{n!} \\ + \end{aligned}\] + + Expanding the series, we get + \[\frac{z^1}{0!} + \frac{z^{-3 + 1}}{1!} + \frac{z^{-6 + 1}}{2!} + ... \] + + Because this series has an infinite number of negative terms, it is an essential singular point. + + \item $\frac{z^3}{1 + z} $ + + First, notice that $f(z)$ has a singularity when $z + 1 = 0 \implies z = -1$. Then, recall the MacLaurin Series + \[ + \frac{1}{1 + z} = \sum^{\infty}_{n=0} (-1)^n z^n + \] + which converges for $|z| < 1$. Therefore, + \[\begin{aligned} + z^3 \cdot \frac{1}{1 + z} + &= z^3 \cdot \sum^{\infty}_{n=0} (-1)z^n & (|z| < 1) \\ + &= \sum^{\infty}_{n=0} (-1)z^{n + 3} \\ + \end{aligned}\] + + By looking at the exponents on $z$ within the sum, we can tell that this series has an infinite number of terms with positive exponents but none with negative ones. Therefore, $z = -1$ is a removal singular point. + \end{enumerate} + + \item[11.] Show that the singular point of each function is a pole and determine its order $m$ and the corresponding residue + + \begin{enumerate} + \item $\frac{z^2 + 2}{z^2 - 1}$ + + \paragraph{Solution} First, notice that this function has two singular points that occur when + \[\begin{aligned} + z^2 - 1 &= 0 \\ + z^2 &= 1 \\ + z &= \pm \sqrt{1} \\ + z &= \pm 1 + \end{aligned}\] + + Also, I will be using the fact that + \[(z - 1)(z + 1) = z^2 - 1\] + + \begin{enumerate} + \item $\mathbf{z = 1}$ + First, rewrite $f(z)$ as follows + \[\begin{aligned} + \frac{z^2 + 2}{z^2 - 1} + &= \frac{z^2 + 2}{(z - 1)(z + 1)} \\ + &= \frac{z^2 + 2}{(z - 1)(z + 1)} \cdot + \frac{\frac{1}{z + 1}}{\frac{1}{z + 1}} \\ + &= \frac{ \mathbf{\frac{z^2 + 2}{z + 1}} }{z - 1} \\ + \end{aligned}\] + + Now, call the bolded part $\phi(z)$. Furthermore, notice that + $\phi(z)$ is analytic at $z = 1$ and that + \[ + \phi(1) = \frac{1^2 + 2}{1 + 1} = \frac{3}{2} \neq 0 + \] + + Therefore, this point is a simple pole with residue $\frac{3}{2}$. + + \item $\mathbf{z = -1}$ + \end{enumerate} + + \item $(\frac{z}{3z + 5})^3$ + \end{enumerate} + + \item[12.] Evaluate the following integrals + +\end{enumerate} +\end{document} \ No newline at end of file diff --git a/Summaries/Common Power Series.pdf b/Summaries/Common Power Series.pdf index 2d2e38062d9f27b4612dd9b2b7ce29f9971088fe..d8299729120942cb83d3284ce104110cf6852e0a 100644 GIT binary patch delta 116388 zcmbrmbzGF)*Dp*rNH@~bF~Gpk(nxm;NOyyhLpMrFNlAl*(j^VjCEX#VGzbXDnHju) z-p}*C&-t9sIe*NWeXVQlwfBnsU2A5z=-7teY)6ZyN8}geOUil144&3sbfOTz^}p2R zc4qr(=G~4oB1OQ!fj}Z}pU)h>Es!cEtwguZ7wn~5pZI^fx;Hi1{szodZMd+*XJKYdi7Ll#($;csR14;r z|Im*rH5yjAg4J5~w*30(sU0uH@{)IBOlt2mmGk6It^ZZ(#FYiOR$m#BSZ9aKeWO&M z3klVTbui__GDy9psRaAES@A~(otTDuiwq*8&Y%N_&&fvILj=@l+IqTeHbwxml7`7? zU&SD@^VuD}4K2GP 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