Question

Find the residues of the following functions, 1. $f(z) = \frac{z^2}{(z-2)(z+1)}$ at the poles $z = 2, -i, i$ 2. $f(z) = \frac{1}{z(z+2)^3}$ at the poles $z = 0, -2$ 3. $f(z) = \frac{ze^{tz}}{(z-3)^2}$ at the poles $z = 3$

          Find the residues of the following functions,
1. $f(z) = \frac{z^2}{(z-2)(z+1)}$ at the poles $z = 2, -i, i$
2. $f(z) = \frac{1}{z(z+2)^3}$ at the poles $z = 0, -2$
3. $f(z) = \frac{ze^{tz}}{(z-3)^2}$ at the poles $z = 3$

Find the residues of the following functions,
1. f(z) = (z^2)/((z-2)(z+1)) at the poles z = 2, -i, i
2. f(z) = (1)/(z(z+2)^3) at the poles z = 0, -2
3. f(z) = (ze^tz)/((z-3)^2) at the poles z = 3

Added by Jeffrey F.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Victor Salazar

09/10/2023

Step 1

We can find the residue at each pole using the formula: $$Res(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z)$$ For $z = 2$, we have: $$Res(f, 2) = \lim_{z \to 2} (z - 2) \frac{z^2}{(z-2)(z^2+1)} = \lim_{z \to 2} \frac{z^2}{z^2+1} = \frac{4}{5}$$ For $z = -i$, we Show more…

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Find the residues of the following functions, 1. $f(z) = \frac{z^2}{(z-2)(z^2+1)}$ at the poles $z = 2, -i, i$ 2. $f(z) = \frac{1}{z(z+2)^3}$ at the poles $z = 0, -2$ 3. $f(z) = \frac{ze^{tz}}{(z-3)^2}$ at the poles $z = 3$