Complex Integration : Find Path using a Transformation

b. Determine the path traced out by w as z moves along a straight line joining A(2 + j0) and B(0 + j2) using the transformation w = z^2. Please see the attached file for the fully formatted problem.

Complex Analysis : Complex Potential for a Fluid, Velocity and Stream Functions

a. A certain fluid is flowing with a constant speed V in a direction making an angle B with the positive x-axis. Find the complex potential for the fluid under consideration. Also determine the velocity and the stream functions.

Complex Analysis : Potential and Electric Field Vector of Two Concentric Charged Cylinders

b. A region is surrounded by two infinitely long concentric cylinders of radii, a1 and a2 (a2>a1). The concentric cylinders are charged to potentials phi1 and phi2 respectively. Determine the potential and electric field vector everywhere in the region. Please see the attached file for the fully formatted problem.

Find Real Roots of an Equation Given Two of the Complex Conjugate Roots

The equation x^4 - 18x^3 + 121x^2 - 368x + 420=0 has complex conjugate roots (4+j2) & (4-j2). By a process of division and solving a quadratic equation, find all the roots and hence write down all the factors of : x^4 ...continues

Confidence Intervals

a. Test is the difference in lifetime of vehicles exist between the two manufacturers at a=0.01. Please see attached.

Analytic Functions; Harmonic Functions; Laplace's Equation

5. Let the function ... be analytic in a domain D that does not include the origin ... 13. ... state why the functions ... are harmonic in D and why ... is in face, a harmonic conjugate 11. ... Why must this satisfy Laplace's equation? Please see attachment for complete questions. Thanks.

Proving ∑ 1/(n^2) = (Pi^2)/6 using Cauchy's Residue Theorem

Deducing that ∑ 1/(n^2) = (Pi^2)/6 is a classic problem in Mathematics. Often it is demonstrated by using Fourier series, but it can also be proved by using Cauchy's Residue Theorem.

Review: Arc, Cauchy-Goursat Theorem, Integral, Paratmetric Representation etc.

Without evaluating the integral show that (see attachment) when C is the same arc as the one in Example 1 (see attachment for example) Please see the attached file for the fully formatted problems.

Review: log Log, Roots, Continuous Complex-Valued Function, Mean Value Theorem etc.

4. By choosing specific nonzero values of z1 and z2, show that ... is not always valid when log is replaced by Log. Please see the attached file for the completely formatted problems. Others are attached.

Evaluating Integrals by Cauchy-Goursat Theorem

Evaluating integrals by Cauchy-Goursat Theorem. See attached for full problem description.