Let f(z) = (4-3z) / ((z-1)(z-2)). Compute f(5)(0). |
|
0027 |
Let f(z) = (4-3z) / ((z-1)(z-2)). According to Laurent's theorem f(z) = sum n = -infinity to infinity of cn(z-1)n for |z-1| > 1. Find ca. |
|
0026 |
Let f(z) = (4-3z) / ((z-1)(z-2)). According to Laurent's theorem f(z) = sum n = -infinity to infinity of cn(z-1)n for 0 < |z-1| < 1. Find ca. |
|
0025 |
Let f(z) = (4-3z) / ((z-1)(z-2)). According to Laurent's theorem f(z) = sum n = -infinity to infinity of cnzn for 1 < |z| < 2. Find ca. |
|
0024 |
Let f(z) = (4-3z) / ((z-1)(z-2)). According to Taylor's theorem f(z) = sum n = 0 to infinity of anzn for |z| < 1. Find aa. |
|
0023 |
Show the integral from 0 to pi of ecos(t)sin(t + sin(t)) with respect to t is 2*sinh(1). |
|
0022 |
Let C be a simple closed contour. Show 1/(2i) * integral(complement(z))dz) = area of the region enclosed by C. |
|
0020 |
Compute an integral along a path. |
|
0019 |
Evaluate an integral along C, where C is different circles in the complex plane. |
|
0018 |
Let z1, z2 be two complex numbers in the third quadrant. Express Log(z1z2) in terms of Log(z1) and Log(z2). |
|
0017 |
Let f(z) = az where a = (-1 + i*3^0.5) / 2. Find all values of Re{f(3/4 - i)}. |
|
0016 |
Let f(z) = az where a = (-1 + i*3^0.5) / 2. For values of z will Re{f(z)} = 0? For what values of z will Im{f(z)} = 0? |
|
0015 |
Solve the equation: ez^2 = (1 - i*3^0.5) / 2. |
|
0014 |
Solve the equation: tan(z) = 2 - i. |
|
0013 |
Show tan-1(z) = i/2 * log((i+z)/(i-z)) for z not equal +/- i. |
|
0012 |
Show tan(z) = tan(x + iy) = (sin(x)cos(x)) / (cos2(x)+sinh2(y)) + i[(sinh(y)cosh(y)) / (cos2(x)+sinh2(y))]. |
|
0011 |
Let U(x,y) = xe-xsin(y) - e-x(y)cos(y). Show U(x,y) is harmonic. Find a V(x,y) such that V(0, pi/2)=0 and f(z) = U(x,y) + iV(x,y) is entire. |
|
0010 |
Let f(z) be an entire function such that: Im{f'(x+iy)} = 12xy-6x, f(1)=-5i, f(i)=-3. Find f(1+i). Express f(z) in terms of z. |
|
0009 |
Let f(z) = f(x+iy)=U(x,y)+iV(x,y) have derivatives of all orders in some domain D. Show the Laplacian of |f(z)|2 is 4|f'(z)|2. |
|
0008 |
Consider the mapping w=z2. Let R by the triangular region in the z-plane with vertices ... Identify and sketch S, the image of R, on the w-plane. |
|
0007 |
Show cos(36) - cos(72) = 1/2. Compute cos(36). Find s, the side of a regular decagon inscribed in a unit circle. |
|
0006 |
The equation: z3 = -11 + 2i has a root z1 = a + bi where a, b are integers. Find, in rectangular form, the three roots of the above equation. |
|
0005 |
The equation z8 = -8 + 8(z^.5)i has two roots, z1 and z2, in the second quadrant. Find in rectangular form z1 and z2. |
|
0004 |
Let z1, z2, z3 be vertices of an equilateral triangle. Show z12 + z22 + z32 = z1z2 + z2z3 + z3z1. |
|
0003 |
Let f(z) = f(reit) = u(r,t) + iv(r,t) have derivatives of all orders in some domain D. Show r2urr + rur + utt = 0 in D. |
|
0002 |
Show: sin(t) + sin(2t) + sin(3t) + ... + sin(nt) = ... |
|
0001 |
Let S be a set of points on the complex plane. Show S is closed if and only if it contains its accumulation points. |
|
0000 |