Evaluate the integral ∫ (log(z) / (z^2 (z - 1)^2)) dz along the closed contour consisting of the line segments from 0 to 2, 2 to 2 + 2πi, and 2 + 2πi to 0, using the Residue Theorem.

Find the value of the integral ∫ (1 / (z^3 (z^2 + 4))) dz, where the contour is the positively oriented circle |z| = 2, using the Residue Theorem and the Cauchy Integral Formula.

Compute the integral ∫ (e^(1/z^2)) / (z^4 + 1) dz along the positively oriented circle |z| = 2, using the Residue Theorem.

In fluid dynamics, the concept of complex potentials is used to study the flow of an incompressible fluid around obstacles. The complex potential Ω(z) for the flow of an ideal fluid around a circular cylinder of radius a, centered at the origin, is given by:

Ω(z) = U (z + a^2 / z)

where U is the free-stream velocity of the fluid, and z = x + iy is the complex variable representing the position in the plane.

If you find the force exerted by the fluid on the circular cylinder by evaluating the complex line integral: F = (1 / (2πi)) ∮_C Ω(z) dz, where the contour C is a circle of radius R > a, centered at the origin, traversed in the counterclockwise direction by devising the complex integral and solving it using the Residue Theorem, the resultant force F on the circular cylinder as: F = 2πi (a^2).

For the differential equation y'' + 2y' + y = e^x cos(x), the complementary solution is y_c = c1e^(-x) + c2e^(-x) cos(x).

If the roots of the characteristic equation for a second-order linear homogeneous differential equation are complex conjugates, then the general solution always involves trigonometric functions.

If F(s) is the Laplace transform of f(t), then the Laplace transform of f''(t) is given by s^2 F(s) -  f(0) - f'(0).

Given a continuous function (f(t)) for which the Laplace transform (L{f(t)}) exists, if (F(s) = L{f(t)}) has a pole at (s = a), then (f(t)) necessarily involves a term proportional to (e^{at}).