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Magazine 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.) $$z^{-3} e^{1 / z^{2}}$$
Problem 5
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. $$\cos z-\sin z$$
Problem 6
Evaluate the following integrals. (Show the details of your work.) $$\int_{0}^{2 \pi} \frac{\sin ^{2} \theta}{5-4 \cos \theta} d \theta$$
Problem 6
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. $$1 /(\cos z-\sin z)$$
Problem 6
Find all the singular points and the corresponding residues. (Show the details of your work.) $$\frac{z^{2}+1}{z^{2}-z}$$
Problem 7
Evaluate the following integrals. (Show the details of your work.) $$\int_{0}^{20} \frac{\cos \theta}{13-12 \cos 2 \theta} d \theta$$. $$\text { Hint .} \cos 2 \theta=\frac{1}{2}\left(z^{2}+\frac{1}{z^{2}}\right)$$
Problem 7
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. $$\frac{\sin 3 x}{\left(x^{2}-1\right)^{2}}$$
Problem 7
Find all the singular points and the corresponding residues. (Show the details of your work.) $$\cot z$$
Problem 8
Evaluate the following integrals. (Show the details of your work.) $$\int_{0}^{2 \pi} \frac{1+4 \cos \theta}{17-8 \cos \theta} d \theta$$
Problem 9
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. $$\cosh \left[1 /\left(z^{2}+1\right)\right]$$
