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Complex Analysis

Cavalieri's Principles Computed

Please see the attached document for homework specifics. Thank-you for your help. Using Cavalieri's principle, compute the volume of the structure...

Derivation of "Minimum Principle" from Maximum Principle

Please see the attached file for the fully formatted problem. The Minimum Principle: Let f be analytic in a bounded region D and continuous and nonzero on bar-D. Show that |f(z)| attains its minimum on the boundary of D. Hint: Consider the function g(z) =1/f(z).

Radicals and complex numbers

Attached 16.) Simplify 17.) Add the complex numbers 2+3i and -3+i 18.) Divide the complex number 2+1 by the complex number 3-i 19.) Factor 20.) Simplify:

Volume of Revolution Problem

Please see the attached file for the complete problem. Find the volume of the solid of revolution formed by rotating the finite region bounded by the graphs of y=sin((pi)x^2)/2) and y=x^5 about the y-axis. The possible answers are: A. 2(5pi-7)/7 B. (7pi-5)/7 C. 2(5-pi)/7 D. (5pi+2)/7 E. 2(7-pi)/7

Polynomial and Contour

Please show all steps to solution. Suppose that P is a polynomial with no roots on the contour Y .Show that the number of roots of P in the region enclosed by Y is given by (see attached file(s).

Parameterized curve integral

Please show all steps to solution (see attached) Let be the curve parameterized by ζ(t) = for Evaluate the integral dz

Euler's equation

Show that y=(A*e^ix)+(B*e^-ix) can be written as y=C*cosine(x-g) A and B are complex but C and g are real Please show all steps!

Image under a map

Please show all logic leading up to answer. Find the image of D = under the map w =

Intermediate value theorem problem

See attached for circled problems Suppose that a function f is continuous on [0,1] except at 0.25 and that f(0)=1 and f(1)=3..... Use the Intermediate Value Theorem to show that there is a root of the given equation on the interval specified.

Applying DeMoivre's Theorem

Please see the attached file for the fully formatted problems. De Moivre's theorem states that (cos theta - i sin theta)^n = cos n theta + i sin n theta for n E R. (a) Use induction to prove de Moivre's theorem for n E Z^+. (b) Show that cos 5 theta = 16 cos^5 theta - 20 cos^3 theta + 5 cos theta. (c) Hence sho

Complex Numbers and Harmonic

1. Find all the values of z in the form a+bi such that (a),(b),(c) (please see the attachment) 2. Find the real part u(x,y) and determine if it is harmonic. (please see the attachment).

Complex analysis for Unit Circle

Let f be analytic inside and on the unit circle. Suppose that 0<|f(z)|<1 if |z| = 1. Show that f has exactly one fixed point inside the unit circle. ( note : a fixed point is a point Zo such that f(Zo) = Zo).

Bijective Conformal Mapping

A) find a bijective conformal mapping that takes a bounded region to an unbounded region b) prove that a conformal map cannot take a simply connected region onto a region that is not simply connected.

DeMoivre's Theorem - Euler Form

Please see the attached file. 1) Express each of the following as a complex number in the Euler form z = r e^ftheta or using the phasor notation z = r <theta [which is an abbreviation for the pola form z = r(cos theta + j sin theta) J: a. ((square root 3) - j)(1 + j(square root 3))) / (1 - j) b. square root(12 - 9j) (pr

An application of Cauchy's inequality

Let f be an entire function such that |f(z)|<=A|z|. Use Cauchy's inequality to show that f(z)=az for some complex constant a. See the attachment for a more complete description of the question and Cauchy's inequality.

Some simple applications of the Cauchy-Goursat theorem

Use the Cauchy theorem to show that the integral around the unit circle |z|=1, traversed in either direction, is zero for each of the following functions: 1) f(z)=z exp(-z) 2) f(z)=tan(z) 3) f(z)=Log(z+2) The attached file contains this question written more clearly with correct mathematical notation.

Green's Theorem: Contour Integrals

** Please see attached file for the complete problem description ** Complex Analysis Problem Problem. Show that if C is a positively oriented simple closed contour, then the area of the region enclosed by C can be written ..... Suggestion: You can use the form: R is a closed region for real valued functions. Does thi

Dividing by Zero and Imaginary Numbers

Mathematicians say that division by zero is forbidden. The expression 5/0, for example, is undefined. "Undefined" in this sense means "unable to be determined". Why is this? When we divide 5 by 5 (5/5) we get 1. Divide 5 by 2 we get 2.5. Each time we make the denominator smaller in the expression 5/x, the expression gets

Determine if the following equations are real or complex.

Determine if the following equations are real or complex; explain the answer in detail. Determine whether the following equations real or complex solutions; justify your answer. Note: It is not necessary to find the solutions; just determine if they are real or complex and explain why. a) 5x2 + 8x + 7 = 0 b) (7)1/2y2 - 6