From 3ad9301f47041b84a90585da1470bcc0b6900ea0 Mon Sep 17 00:00:00 2001 From: vincentlaucsb Date: Wed, 23 Aug 2017 11:10:46 -0700 Subject: [PATCH] Added Stuff on Principal Part --- ...ework 6 - Principal Part of a Function.pdf | Bin 0 -> 120486 bytes ...ework 6 - Principal Part of a Function.tex | 133 ++++++++++++++++++ Summaries/Common Power Series.pdf | Bin 0 -> 29504 bytes Summaries/Common Power Series.tex | 15 ++ ...emovable and Essential Singular Points.tex | 69 +++++++++ Summaries/Poles.pdf | Bin 0 -> 145730 bytes 6 files changed, 217 insertions(+) create mode 100644 122B Homework 6 - Principal Part of a Function.pdf create mode 100644 122B Homework 6 - Principal Part of a Function.tex create mode 100644 Summaries/Common Power Series.pdf create mode 100644 Summaries/Common Power Series.tex create mode 100644 Summaries/Poles, Removable and Essential Singular Points.tex create mode 100644 Summaries/Poles.pdf diff --git a/122B Homework 6 - Principal Part of a Function.pdf b/122B 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following functions at the singular point and determine whether that point is a pole, a removable singular point, or an essential singular point: + \begin{enumerate} + \item $\frac{z^2}{1 + z}$ + \paragraph{Solution} + + This function has a singular point at $z = -1$. + First, we will find the Laurent series expansion. Notice that, + \[\begin{aligned} + z^2 + &= [(z + 1)^2 - 2z - 1] \\ + &= [(z + 1)^2 - 2(z + 1) + 1] \\ + \end{aligned}\] + + Therefore, + \[ + \begin{aligned} + \frac{z^2}{1 + z} + &= z^2 \cdot \frac{1}{1 + z} \\ + &= [(z + 1)^2 - 2(z + 1) + 1] \cdot \frac{1}{1 + z} \\ + &= \frac{(z + 1)^2 - 2(z + 1) + 1}{1 + z} \\ + &= (z + 1) - 2 + \frac{1}{1 + z} + \end{aligned} \] + + By inspection, the principal part of this function is + \[\frac{1}{1 + z}\] + so $z = -1$ is a pole of order 1. + + \item $\frac{\sin{z}}{z}$ + \paragraph{Solution} + First, we will try to find a Laurent series representation for $f(z)$. + + \[\begin{aligned} + \frac{\sin{z}}{z} + &= \frac{1}{z} \cdot \sin{z} \\ + &= \frac{1}{z} \cdot \sum^{\infty}_{n=0} (-1)^n \frac{z^{2n + 1}}{(2n + 1!)} + & (|z| < \infty) \\ + &= \frac{1}{z} \cdot [(-1)^0 \cdot \frac{z^1}{1!} + + (-1)^1 \frac{z^3}{3!} + (-1)^2\frac{z^5}{5!} + ...] & \text{Expanding the series} \\ + \end{aligned}\] + + By inspection, this Laurent Series has no terms with negative exponents and is really just a regular power series. Therefore, it is a removable singular point. + \end{enumerate} + + \item Show that the singular point of each function is a pole and determine its order $m$ and find the corresponding residue: + \begin{enumerate} + \item $(\frac{z}{2z + 1})^3$ + \paragraph{Solution} + First, this function has singularities whenever $(2z + 1)^3 = 0$, implying $z = -\frac{1}{2}$ is a singular point. + + Now, notice that + \[\begin{aligned} + \frac{z}{2z + 1} + &= \frac{z}{2} \cdot \frac{1}{z + \frac{1}{2}} \\ + \end{aligned}\] + + Furthermore, because $z = (z + \frac{1}{2}) - \frac{1}{2}$, the above is equivalent to + \[\begin{aligned} + \frac{z}{2z + 1} + &= \frac{(z + \frac{1}{2}) - \frac{1}{2}}{2} \cdot + \frac{1}{z + \frac{1}{2}} \\ + &= \frac{1}{2} \cdot \frac{ + (z + \frac{1}{2}) - \frac{1}{2}}{ + z + \frac{1}{2}} \\ + &= \frac{1}{2} \cdot [1 - \frac{0.5}{z + \frac{1}{2}}] \\ + \end{aligned}\] + + Finally, we get + \[\frac{1}{2} - \frac{0.25}{z + \frac{1}{2}}\] + + Therefore, because there is only $(z - z_0)^n$ term with an exponent of $-1$, this is a pole of order 1 with residue 0.25. + + \item $\frac{\exp{z}}{z^2 + \pi^2}$ + \paragraph{Solution} + Consider $z = \pi i$. At this point, the denominator $z^2 + \pi^2 = \pi^2 \cdot i^2 + \pi^2 = 0$, so $\pi i$ is a singular point. + + \bigskip + + First, let us find the Taylor Series of $\exp(z)$ about $z = \pi i$. Because + $\frac{d}{dz} \exp(z) = \exp(z)$, and $\exp(\pi i) = -1$, + \[\begin{aligned} + \frac{f^{(n)}(z_0)}{n!} + &= \frac{\exp(\pi i)}{n!} \\ + &= \frac{-1}{n!} \\ + \end{aligned}\] + + Thus, the Taylor Series of $\exp(z)$ centered at $\pi i$ is + \[ f(z) = \sum^{\infty}_{n=0} \frac{-1}{n!} (z - \pi i)^n \] + which converges for all $z \in \mathbb{C}$. + + Notice that + \[(z + \pi i)(z - \pi i) + = (z^2 - z\pi i + z\pi i - \pi^2 i^2) + = z^2 + \pi^2 \] + + Therefore, + \[\begin{aligned} + \frac{\exp{z}}{z^2 + \pi^2} + &= \frac{1}{(z + \pi i)(z - \pi i)} \cdot + \sum^{\infty}_{n=0} \frac{-1}{n!} (z - \pi i)^n \\ + &= \frac{1}{z + \pi i} \cdot + \sum^{\infty}_{n=0} \frac{-(z - \pi i)^{n - 1}}{n!} \\ + \end{aligned}\] + + And the completion of this solution has been left as an exercise for the reader. + \end{enumerate} +\end{enumerate} +\end{document} \ No newline at end of file diff --git a/Summaries/Common Power Series.pdf b/Summaries/Common Power Series.pdf new file mode 100644 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