Math 136: Complex Variables and Integral Transforms

Homework 11

Read Sections 6.5, 6.6, 7.1, 7.2 of Saff and Snider,
and the summary at the end of Chapter 6.

Do these problems from Saff and Snider:

  • Fun Problem (A) (1 point): What contour would you use to evaluate the integral from 0 to infinity of ( x300 + a300 )-1, where a > 0?
    (Don't evaluate it!)
  • Sec. 6.4, Improper Integrals, Jordan's Lemma: 6, 12. In 12, also take the real and imaginary parts of the answer, in order to find the values of the integrals of cos(x^2) and sin(x^2) from 0 to infinity. These are known as the Fresnel integrals.

    Hint for Q12 The hard part is to show that the integral over the circular arc in your contour tends to zero. Here you need to write out the function e^{iz^2} fully, and use the fact that sin(2 theta) is greater than or equal to (4/pi)theta, for all theta in [0, pi/4]. Make sure to explain why this fact is true. In general, the proof is a lot like the proof of Jordan's Lemma in the text, and uses Lemma 2 of Section 6.4 in the same way.
  • Sec. 6.5, Indented Contours: 1(b), 4.
    (Think about 5, 12.)
  • Sec. 6.6, Multiple-Valued Functions: 2.
    (Think about 10, 11, 12. For 10, you would need to verify the estimates over C_epsilon and C_R. Why is the note in Q8 true?)

Few problems this week, but most require stamina. Use scrap paper.
Be sure to explain your answers fully, including estimates of integrals over C_R going to zero (where needed).

Happy Thanksgiving!

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