Phys 2920, Spring 2009
Problem Set #9
1. Use a computer program which does “3D graphics”, make a plot of the function
f(x, y) = 5 sin(2x) sin(5y)
for the range 0 ≤x≤2π, 0 ≤y≤2π.
Hint: In Maple, you can learn how the procedure plot3d is used. You may also want to change
the option numpoints or grid to get a better picture.
2. Using a computer program produce (any way you can) a plot of the function
f(ρ, φ) = 5 sin(ρ2/3) cos(φ)e−ρ2/10
for the range −5≤x≤5, −5≤y≤5
3. Find cos(−2 + 5i), using both your calculator and demonstrating the result using only a real-
variable ($10) calculator.
4. Evaluate cos−1(10); get the decimal value from your calculator and show how this value comes
about using the definition of cos−1(z) and real-variable math.
5. (CV 3.44) Prove that d
dz (z2z∗) does not exist anywhere.
(If it were analytic, the C–R equations would have to be satisfied...)
6. (CV 3.47) Verify that the real and imaginary parts of the following functions satisfy the Cauchy–
Riemann equations and thus deduce the analyticity of each function:
(a)f(z) = z2+ 5iz + 3 −i , (b)f(z) = ze−z,(c)f(z) = sin z
In other words, use z=x+iy, find u(x, y )and v(x, y)and test the C–R equations.
7. (CV 3.78) Evaluate
(a) lim
z→2i
z2+ 4
2z2+ (3 −4i)z−6i(b) lim
z→eπi/3(z−eπi/3)z
z3+ 1(c) lim
z→i
z2−2iz −1
z4+ 2z2+ 1
(A little practice with l’Hopital’s rule and derivatives.)
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