Cauchy's Theorem and Deforming a Contour · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Why closed loops often integrate to zero

Mathematics · Complex Analysis · ages 20-21
Name ______________________   Date ____________

  1. Deforming a contour across a region where f is analytic leaves the integral unchanged. If the integral over the original loop equals 3 + 4, what is it over the deformed loop?

    Answer: ______________

  2. The integral of 1 divided by z around the unit circle is nonzero. Why does this not contradict Cauchy's theorem?

    • The unit circle is not a smooth contour
    • 1 divided by z has a singularity at 0 inside the loop, so the theorem does not apply
    • 1 divided by z is not continuous at points of the circle
    • Cauchy's theorem only applies to real integrals

  3. Green plus Cauchy-Riemann zeroes both the real and imaginary parts of a closed loop integral for analytic f.

    Circle one:   True   False

  4. Write f equal u plus iv. What is the real part of the loop integral of f dz?

    • u dx minus v dy
    • u dx plus v dy
    • v dx minus u dy

  5. Two loops differ by deformation across a region where f is analytic. The integral over the small loop is 1.5 + 1. What is the integral over the large loop?

    Answer: ______________

  6. f is analytic inside and on a loop except at one interior point where it blows up. The loop integral must still be zero.

    Circle one:   True   False

  7. Write f equal u + iv. The real part of the loop integral of f dz equals which real line integral?

    • The integral of u dx minus v dy
    • The integral of u dx plus v dy
    • The integral of v dx plus u dy
    • The integral of u dy minus v dx

  8. Suppose f is analytic inside and on a loop except at one interior point where it blows up. The loop integral must still be zero. True or false?

    Circle one:   True   False

  9. At a point, u_x is 3, u_y is 4, v_x is minus 4 and v_y is 3. With P equal u and Q equal minus v, what is Q_x minus P_y?

    Answer: ______________

  10. The integral of z squared around any closed loop is 0, but the integral of 1 divided by z around the unit circle is 2 pi i. What is the essential difference?

    • z squared is a polynomial of even degree
    • The unit circle is a contour made for 1 divided by z
    • z squared is analytic everywhere while 1 divided by z has a singularity inside the loop
    • 2 pi i equals 0 in complex arithmetic
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Answer key

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Why closed loops often integrate to zero W1-mt_aacYFVN-HH-s1

  1. 7 · Deformation preserves the value, and 3 + 4 is 7.
  2. 1 divided by z has a singularity at 0 inside the loop, so the theorem does not apply · Cauchy's theorem needs analyticity everywhere inside, but 1 divided by z blows up at the interior origin.
  3. True · Each part becomes a circulation whose curl is a Cauchy-Riemann difference.
  4. u dx minus v dy · Multiplying out (u plus iv)(dx plus i dy) gives that real part.
  5. 2.5 · Deformation preserves integrals, and 1.5 + 1 is 2.5.
  6. False · One interior singularity can produce a nonzero integral, as 1 divided by z shows.
  7. The integral of u dx minus v dy · Multiplying out (u + iv)(dx + i dy) gives real part u dx minus v dy.
  8. False · False. A single interior singularity can produce a nonzero integral, as 1 divided by z shows.
  9. 0 · Q_x is 4 and P_y is 4, and their difference is the Cauchy-Riemann cancellation.
  10. z squared is analytic everywhere while 1 divided by z has a singularity inside the loop · Entire functions integrate to zero on loops. The pole of 1 divided by z inside is the whole story.
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