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題組內容

Problem 7. Let f(&) be a function of a complex variable x = a + iy.

(a) Assume that Re{f(z)} = x³ -3xy². Find the imaginary part Im{f(z)} so that f(z) is analytic for all x .

其他申論題

(b) the second-order ODE:

(b)f(x)= {1,-1≤x≤1, 0,x＜-1;x＞1

Problem 6. Use Laplace transform to solve the following differential equation:

(b) Assume that u(x, y) = x3 +3xy2. Show that it is impossible to find a real-valued function v(x, y) such that f(z) = u(x,y) + iu(x,y) is differentiable with respect to z for all z .

(a) Calculate f'(0) and f"(0).

(b) Find the Taylor series expansion of f(z) about the point z = 0; that is, find the coefficients such that in a certain neighborhood of x = 0.

(c) Continuing from part (b), what is the region of convergence?

(d) Let C denote the path along the y axis from -∞ to oo. Calculate f(z)dz; i.e., evaluate the following integral:

(一)、Describe the harmonic problems of a Y-Y connected three-phase transformer. (4%)

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