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8.4. Homework Problems

1. Find arg i, arg(-i), and arg 37, where arg z denotes the argument of the complex number z.

2. Visualize the following multiplications in the complex plane using the interpretation of multiplication in terms of multiplying magnitudes and adding arguments: (i)(i) = -1, (i)(-i) = 1, (-i)(-i) = -1.

3. If we visualize z as a point in the complex plane, how should we visualize -z?

4. Find four different complex numbers z such that z4 = 1.

5. Compute the following:

\begin{align} |1+i| , \arg(1+i) , \left|\frac{1}{1+i}\right| , \\ \arg\left(\frac{1}{1+i}\right) , \frac{1}{1+i} \end{align}

6. Write the function tan x in terms of complex exponentials.

7. Evaluate \(\int \sin^3 xdx\).

8. Use Euler's theorem to derive the addition theorems that express sin(a+b) and cos(a+b) in terms of the sines and cosines of a and b. (solution in the pdf version of the book)

9. Evaluate

\[ \int_0^{\pi/2} \cos x \: \cos 2x dx . \]

(solution in the pdf version of the book)

10. Find every complex number z such that z3 = 1. (solution in the pdf version of the book)

11. Factor the expression x3-y3 into factors of the lowest possible order, using complex coefficients. (Hint: use the result of problem 10.) Then do the same using real coefficients. (solution in the pdf version of the book)

12. Evaluate

\[ \int \frac{dx}{x^3-x^2+4x-4} . \]

13. Evaluate

\[ \int e^{-ax}\cos bx dx . \]

14. Consider the equation f'(x) = f(f(x)). This is known as a differential equation: an equation that relates a function to its own derivatives. What is unusual about this differential equation is that the right-hand side involves the function nested inside itself. Given, for example, the value of f(0), we expect the solution of this equation to exist and to be uniquely defined for all values of x. That doesn't mean, however, that we can write down such a solution as a closed-form expression. Show that two closed-form expressions do exist, of the form f(x) = axb, and find the two values of b. (solution in the pdf version of the book)

15. (a) Discuss how the integral

\[ \int \frac{dx}{x^{10000}-1} \]

could be evaluated, in principle, in closed form. (b) See what happens when you try to evaluate it using computer software. (c) Express it as a finite sum. (solution in the pdf version of the book)