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Magazine Problem 1
Expand the given function in a Laurent series that converges for \(0< |z|< R\) and determine the precise region of convergence, (Show the details of your work.) $$\frac{1}{z^{4}-z^{5}}$$
Problem 2
Determine the location and kind of the singularities of the following functions in the finite plane and at infinity, In the case of poles also state the order. $$z+\frac{2}{z}-\frac{3}{z^{2}}$$
Problem 2
Evaluate the following integrals. (Show the details of your work.) $$\int_{0}^{\infty} \frac{d \theta}{2+\cos \theta}$$
Problem 2
Expand the given function in a Laurent series that converges for \(0< |z|< R\) and determine the precise region of convergence, (Show the details of your work.) $$z \cos \frac{1}{z}$$
Problem 3
Evaluate the following integrals. (Show the details of your work.) $$\int_{0}^{2 \pi} \frac{d \theta}{37-12 \cos \theta}$$
Problem 3
Find all the singular points and the corresponding residues. (Show the details of your work.) $$\frac{1}{4+z^{2}}$$ .
Problem 3
Determine the location and kind of the singularities of the following functions in the finite plane and at infinity, In the case of poles also state the order. $$\cot z^{2}$$
Problem 4
Expand the given function in a Laurent series that converges for \(0< |z|< R\) and determine the precise region of convergence, (Show the details of your work.) $$\frac{\cosh 2 z}{z^{2}}$$
Problem 4
Evaluate the following integrals. (Show the details of your work.) $$\int_{0}^{20} \frac{d \theta}{8-2 \sin \theta}$$
Problem 5
Expand the given function in a Laurent series that converges for \(0< |z|< R\) and determine the precise region of convergence, (Show the details of your work.) $$z^{-3} e^{1 / z^{2}}$$
