Complex Variables

2. Suppose that f(z) is entire and that the harmonic function... Show that u(x,y) must be constant throughout the plane... See the attachment for all problems.

Use Residues to Evaluate Integrals (3 Problems)

Please see the attached file for the fully formatted problems.

Complex Variables : Substitution and Beta Function

See the attachment (77-7) for the problem. and you should also see the examples in section 77 in order to solve it. I also attached section 77 (example1-1 and example1-2).

Residue calculus

Use residues to compute... see the attachment for the problems. Thank you

Four complex variable questions (83-5, 85-5, 7, 13)

see the attachment for the problems. --- - Find the region onto which the half plane y>0... - Find the image of the quantrant x>1, y>0... - Describe geometrically the transformation w=1/(z-1). - Using the exponential formz = re^(i x theta) of z... ---- (See attached file for full problem description)

Four complex variable problems (87-4,5,6,10)

see the attachment for the problems --- - Find the bilinear transformation that maps... - Show the disposition of two linear fractional transformations... - A fixed point of transformation w = f(z)... - Show that there is only linear fractional transformation that maps... --- (See attached file for full problem descri ...continues

complex variables (88-1, 3, 89-3, 5)

See the attachment for the problems --- - transformation maps the half plane Im z>0... - finding the inverse of the transformation... - Vertical half lines... - Verify that the interior of a rectangular region... --- (See attached file for full problem description)

Two complex variable problems (90-4,7) two problems

see the attachment for the problems. --- - the image of the closed rectangular region... - principal branch of the square root... --- (See attached file for full problem description)

Complex Variables : Analytic Funcitons and Contours

Please see the attached file for the fully formatted problems. --- - Lef f be a function which is analytic inside... - Write and equation... --- (See attached file for full problem description)

Complex Variables : Analytic Functions, Contours, Polynomials and Rouche's Theorem

Please see the attached file for the fully formatted problems. --- - Suppose the function f is analytic inside... - Determine the number of zeros... - Write f(z) - z^n and... - Any polynomial... --- (See attached file for full problem description)