## Question

z⁴ + 2*(z²) + 2 = 0

2. Establish the following result:

z - conjugate(z) = 2*i*Im(z)

3. Sketch the regions associated with the following inequalities. Determine if the region is open, closed, bounded, or compact.

Re(z) >= 4

4. Sketch the following regions. Determine if they are connected, and what the closure of the region is if they are not closed.

0 <= arg(z) < 2*pi

5. Discuss the following transformations (mappings) from the z plane to the w plane; here z is the entire finite complex plane.

w = 1/z

6. Relating to the unit circle and sterographic projections, to what curves on the sphere do the lines Re(z) = x = 0 and Im(z) = y = 0 correspond?

7. Evaluate the following limit:

Limit as z goes to infinity of: sin(z)/z

8. Evaluate the following limit:

Limit as z goes to infinity of: z/((z²) + 1)

9. Where is the following function differentiable?

tan(z)

10. Let z = x be real. Use the relationship (d/dx)eᶦˣ = i*eᶦˣ to find the standard derivative formulae for trigonometric functions:

(d/dx)sin(x) = cos(x)

(d/dx)cos(x) = -sin(x)

11. Suppose we are given the following differential equation:

((d³)w)/(d(t³)) - (k³)w = 0

Where t is real and k is a real constant. Find the general real solution of the above equation. Write the solution in terms of real functions.

12. In the following we are given the real part of an analytic function of z. Find the imaginary part and the function of z.

3*(x²)*y - y³

13. Determine whether the following functions are analytic. Discuss whether they have any singular points or if they are entire.

tan(z)

e⁽¹/⁽ᶻ⁻¹⁾⁾

14. Evaluate the closed contour integral of f(z)dz, where C is the unit circle enclosing the origin, and f(z) is given as follows:

z*conjugate(z)

15. Let C be the unit square with diagonal corners at -1 -i and 1 + i. Evaluate the closed contour integral of f(z)dz, where f(z) is given by the following:

1/(2*z + 1)

16. Use the principal branch of log(z) and z¹/² to evaluate:

The Integral from -1 to 1 of log(z)dz

17. (Cauchy's Theorem) Use partial fractions to evaluate the following integrals (Closed Contour integral of f(z)dz), where C is the unit circle centered at the origin, and f(z) is given by the following:

1/(z*(z - 2))

18. (Cauchy's Integral Formula) Evaluate the closed contour integrals of f(z)dz, where C is the unit circle centered at the origin and f(z) is given by the following (use the power series representations of e, sin, sinh, cos, and cosh as necessary):

(a) 1/((2*z - 1)³)

(b) (e⁽ᶻ²⁾)*(1/(z²) - 1/(z³))

19. (Cauchy's Integral Formula) Evaluate the closed contour integrals f(z) dz over a contour C, where C is the boundary of a square with diagonal opposite corners at z = -(1 + i)R and z = (1 + i)R, where R > a > 0, and where f(z) is given by the following (use the power series representations mentioned in problem 18 as necessary):

(eᶻ)/(z - (pi*i/4)*a)

20. (Cauchy's Integral Formula) Evaluate the integral from negative infinity to positive infinity of 1/((x + i)²)dx by considering the closed contour integral C(R) (1/((z + i)²))dz, where C(R) is the closed semicircle in the upper half plane with corners at z = -R and z = R, plus the x axis. Hint: show that

limit as R->inf of the integral on the contour C1(R) (1/((z + i)²))dz = 0 where C1(R) is the open semicircle in the upper half plane (not including the x axis).

21. Write each of the following expressions below in the form a + ib. If there are multiple values, state them all and then give the principal value. Show all work and simplify your answers:

(a) sinh(pi + i)

(b) (1 + i)¹/³

(c) i²ᶦ

22. For the following functions:

(1) Determine all of the singular points in the finite complex plane

(2) Determine the type of each singular point, and if it is a pole, state its order

(3) Determine the residue at each pole.

(a) f(z) = sinh(z)/(z³)

(b) f(z) = 1/(z*cos(z))

23. Determine the Taylor series of the function:

f(z) = (1 - cosh(z²))/(z²)

about z = 0 and state the radius of convergence.

24. Determine the Laurent series of the function:

f(z) = z/((z²) - 4)

about z = 0 that is valid in the region 2 < |z|. Hint: The series you obtain should be in powers of z.

25. Use contour integration to compute the following improper integral:

Integral from 0 to positive infinity of 1/((1 + x²)²)dx

Justify all your steps and simplify your answers.

26. (Fourier-type integral) Use contour integration to compute the following improper integral:

Integral from 0 to positive infinity of (x*sin(2x))/((x²) + 4)dx

Justify all your steps and simplify your answers.

27. (Laplace Transform) Use contour integration on the Bromwich contour to compute the inverse Laplace transform of

F(s) = 2/(s((s²) + 1))

Show all your steps.

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