# Untitled Page 35

- Page ID
- 148379

## 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 *z*^{4}=1.

**5**.
Compute the following:

**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

(solution in the pdf version of the book)

**10**.
Find every complex number *z* such that *z*^{3}=1.
(solution in the pdf version of the book)

**11**.
Factor the expression *x*^{3}-*y*^{3} 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

**13**.
Evaluate

**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*)=*ax*^{b}, and find the two values of *b*.
(solution in the pdf version of the book)

**15**.
(a) Discuss how the integral

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)

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.

### Footnotes

*z*'s argument is 30 degrees, then we could say z

^{-}'s was -30, but we could also call it 330. That's OK, because 330+30 gives 360, and an argument of 360 is the same as an argument of zero.