Question

Q4. (a) Determine the Laurent expansion of $f(z) = \frac{1}{z^2 + z - 6}$ valid in the annulus $A = \{z \in \mathbb{C} : 2 < |z| < 3\}$. (b) State Cauchy's Residue theorem and explain what is meant by the term residue. (c) Let $g(z) = \frac{z^2}{4 + z^4}$. Where are the poles of $g$ and what is their order? (See Q1(f).) Calculate the residues of $g$ at any poles which lie in the upper half plane. (d) Evaluate the improper real integral $\int_0^\infty \frac{x^2}{4 + x^4} dx$. You may like to use a semi-circular contour in the upper half-plane and appeal to part (c).

          Q4. (a) Determine the Laurent expansion of $f(z) = \frac{1}{z^2 + z - 6}$ valid in the annulus $A = \{z \in \mathbb{C} : 2 < |z| < 3\}$.
(b) State Cauchy's Residue theorem and explain what is meant by the term residue.
(c) Let $g(z) = \frac{z^2}{4 + z^4}$. Where are the poles of $g$ and what is their order? (See Q1(f).) Calculate the residues of $g$ at any poles which lie in the upper half plane.
(d) Evaluate the improper real integral $\int_0^\infty \frac{x^2}{4 + x^4} dx$. You may like to use a semi-circular contour in the upper half-plane and appeal to part (c).

$Q4. (a) Determine the Laurent expansion of f(z) = (1)/(z^2 + z - 6) valid in the annulus A = {z ∈ℂ : 2 < |z| < 3}. (b) State Cauchy's Residue theorem and explain what is meant by the term residue. (c) Let g(z) = (z^2)/(4 + z^4). Where are the poles of g and what is their order? (See Q1(f).) Calculate the residues of g at any poles which lie in the upper half plane. (d) Evaluate the improper real integral ∫0^∞(x^2)/(4 + x^4) dx. You may like to use a semi-circular contour in the upper half-plane and appeal to part (c).$

Added by Rafael A.

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