Evaluate the integral ∫ (z^2 + 2z + 3) / (z^3 - 1) dz along the circle |z| = 2 in the counterclockwise direction using the Residue Theorem.

Find the value of the integral ∫ (e^(z^2)) / (z^4 + 1) dz along the positively oriented square contour with vertices at ±1 ± i.

Compute the integral ∫ (sin(z) / (z^3 (z^2 + 1))) dz, where the contour is the positively oriented circle |z| = 3, using the Residue Theorem.

In potential theory, the concept of complex potentials is used to study the behavior of electric and gravitational fields. The complex potential Φ(z) for a point charge q located at the origin in a two-dimensional plane is given by: Φ(z) = (q / (2π)) log(z), where z = x + iy is the complex variable representing the position in the plane. Suppose we have a system of two point charges, q1 and q2, located at z1 and z2, respectively, in the complex plane. The complex potential Ω(z) for this system is the sum of the individual potentials:

Ω(z) = (q1 / (2π)) log(z - z1) + (q2 / (2π)) log(z - z2).

If your task is to find the work done in moving a unit positive test charge once around a closed circular path C of radius R, centered at the origin, in the counterclockwise direction & this work is given by the complex line integral: W = ∮_C Ω(z) dz, then the work done in moving a unit positive test charge once around the circular path C in the counterclockwise direction obtained by you is : W = i (q1 + q2).

If the roots of the characteristic equation for a second-order linear homogeneous differential equation are real and distinct, then the general solution always contains exponential terms.

The particular solution of the differential equation y'' - 6y' + 9y = x^3 is y_p = x^3/18.

If f(t) is a piecewise continuous function with f(t) = 0 for t < 0, then the Laplace transform of f(t) exists for all values of s.

The Laplace transform of the derivative of a function is equal to s times the Laplace transform of the function minus the value of the function at t = 0.