Find Expressions for Bounded Steady State Temperatures and Isotherms For a Plate and Solve a Dirichlet problem for a Semi-infinite Strip

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Solve a Dirichlet Equation for a Semi-infinite Strip; Cosine Series and Poisson Kernel and a Complex Summation

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Complex Variables : Stream Functions and Flow Around a Corner

(See attached file for full problem description) --- - flow around a corner... - components of velocity... --- (See attached file for full problem description)

Complex Variables : Region of Flow, Fluid Presure and Speed of Fluids

4. Show that the speed of the fluid at points on the cylindrical surface in Example 2, Sec. 108. (below) is 2A| sin θ| and also that the fluid pressure on the cylinder is greatest at the points z = ±1 and least at the points z = ± i --- - Show the speed of the fluids... - At an interior point of a region of flow... -- ...continues

Complex Variables : Find the image of the semi-infinite strip x > 0, 0 < y < 2 when w = iz + 1. Sketch the strip and its image.

Find the image of the semi-infinite strip x > 0, 0 < y < 2 when w = iz + 1. Sketch the strip and its image. --- Please see the attached file for the fully formatted problems.

Functions : Analyticity and Differentiability

Assume that f(z) is analytic at the origin and f(0) = first derivative of f at 0 = 0. Prove that f(z) can be written in the form f(z) = [z^2]g(z), where g(z) is analytic at z = 0.

Prove a function is analytic in an open disk.

Let g be continuous on the real interval [0,1] and define H(z) := integral (from 0 to 1) [g(t)/(1-z[t^2])]dt, (|z| < 1) Prove that H is analytic in the open disk |z| < 1.

Does the principal branch square root of z have a Laurent series expansion in the domain C{0}?

Does the principal branch square root of z have a Laurent series expansion in the domain C{0}? Explain.

Laurent series for a trigonometric function : (z^2)*cos(1/(3z)) in |z| > 0

Find the Laurent series for (z^2)*cos(1/(3z)) in |z| > 0

Derivatives and Application of the Derivative

f(t) = 10,000 / 10 + 50e ^-0.5t HOW do I obtain the derivative? What is the "e" portion of the problem? I know the derivative = 250,000e^-0.5t/ (10+50e ^-0.5t)^2 Please describe in detail the steps taken to arrive at this answer. For example, Why is the top of the equation 250,000e^-0.5t? Why is the bottom (10+50e^-0 ...continues