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Untitled Page 35

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  • 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:




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

    7. Evaluate \int sin3 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

    2} cos x : cos 2x dx qquad .

    (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


    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)=axb, 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.


    [1] I cheated a little. If 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.
    [2] In general, the use of complex number techniques to do an integral could result in a complex number, but that complex number would be a constant, which could be subsumed within the usual constant of integration.
    [3] See page 151 for an explanation of where this definition comes from and why it makes sense.