Question

Q.13. Find the Laurent series expansion of the following functions about the origin. (a) $f(z) = \sin(\frac{1}{z^2})$ (d) $f(z) = e^{z^2} + e^{\frac{1}{z^2}}$ (b) $f(z) = \frac{e^z}{\sin z}$ (e) $f(z) = z^k \sin(\frac{1}{z})$, where $k \in \mathbb{N}$. (c) $f(z) = e^{\frac{1}{z^2}}$

          Q.13. Find the Laurent series expansion of the following functions about the origin.
(a) $f(z) = \sin(\frac{1}{z^2})$
(d) $f(z) = e^{z^2} + e^{\frac{1}{z^2}}$
(b) $f(z) = \frac{e^z}{\sin z}$
(e) $f(z) = z^k \sin(\frac{1}{z})$, where $k \in \mathbb{N}$.
(c) $f(z) = e^{\frac{1}{z^2}}$

Q.13. Find the Laurent series expansion of the following functions about the origin.
(a) f(z) = sin((1)/(z^2))
(d) f(z) = e^z^2 + e^(1)/(z^2)
(b) f(z) = (e^z)/(sin z)
(e) f(z) = z^k sin((1)/(z)), where k ∈ℕ.
(c) f(z) = e^(1)/(z^2)

Added by Mary C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

04/04/2023

Step 1

Step 1: The Laurent series expansion of a function f(z) about a point z = a is given by: $$f(z) = \sum_{n=-\infty}^{\infty} c_n (z-a)^n$$ where the coefficients $c_n$ are given by: $$c_n = \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z-a)^{n+1}} dz$$ where C is a closed Show more…

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Q.13. Find the Laurent series expansion of the following functions about the origin. (a) f(z) = sin($\frac{1}{z^2}$) (d) f(z) = $e^{z^2}$ + $e^{(\frac{1}{z^2})}$ (b) f(z) = $\frac{e^z}{sin z}$ (e) f(z) = $z^k$ sin($\frac{1}{z}$), where k ∈ N. (c) f(z) = $e^{(\frac{1}{z})}$