Contour Integrals Along Parametrised Paths · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Walking a function along a path

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

  1. With gamma(t) equal t for t in [0, 3], the contour integral of z dz becomes the real integral of t dt from 0 to 3. What is its value?

    Answer: ______________

  2. Parametrise the upper half of the unit circle from 1 to minus 1 by gamma(t) equal e to the power it with t in [0, pi]. What is the integral of 1 dz along gamma?

    • 2
    • 0
    • pi times i
    • minus 2

  3. With gamma of t equal t for t in 0 to 3, the integral of z dz becomes the integral of t dt. What is its value?

    Answer: ______________

  4. Reversing the direction of a contour changes the sign of the contour integral.

    Circle one:   True   False

  5. A contour has length 3 and |f| is at most 4 on it. What bound does the estimation lemma give for the absolute value of the integral?

    Answer: ______________

  6. Which change leaves a contour integral unchanged?

    • Travelling the same path backwards
    • Travelling the same path twice as fast
    • Keeping the speed but changing the path

  7. Let gamma(t) equal e to the power it with t in [0, 2 pi]. What is the integral of 1 divided by z dz around gamma?

    • 0
    • 2 pi
    • 2 pi i
    • minus 2 pi i

  8. Two parametrisations trace the same circle, one clockwise and one anticlockwise. Their contour integrals of the same function are equal. True or false?

    Circle one:   True   False

  9. On the unit circle f is at most 2. Using 6.28 for the circumference, what bound does the estimation lemma give?

    Answer: ______________

  10. Let gamma wind once anticlockwise around the circle of radius 1 centred at 2. What is the integral of 1 divided by z dz around gamma?

    • 0
    • 2 pi i
    • 1
    • 4 pi i
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Walking a function along a path W1-mt_CaJ7ruAv8P-s1

  1. 4.5 · The antiderivative is t squared over 2, giving 9 divided by 2, which is 4.5.
  2. minus 2 · Integrating 1 just gives end minus start: minus 1 minus 1 equals minus 2.
  3. 4.5 · The antiderivative is t squared over 2, giving 9 over 2.
  4. True · Traversing backwards negates the velocity, hence the whole integral.
  5. 12 · Maximum modulus times length: 4 times 3 equals 12.
  6. Travelling the same path twice as fast · Direction preserving reparametrisation compensates through the velocity factor.
  7. 2 pi i · On the unit circle 1 divided by z equals e to minus it, and the velocity e to it cancels it, leaving the integral of i, which is 2 pi i.
  8. False · False. Opposite directions give opposite signs, so the integrals are negatives of each other.
  9. 12.56 · Twice 6.28 equals 12.56.
  10. 0 · The disc spans real parts 1 to 3, which excludes the origin, so 1 divided by z is analytic inside and the integral is 0.
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