For γ(t) = t + it² on [0, 1], what is the imaginary part of ∫_γ z dz?

For γ(t) = t + it on [0, 1], what is γ'(t)?

• 1 + i
• √2
• 1

With γ(t) = t for t in [0, 3], the integral of z dz becomes the integral of t dt. What is its value?

Reversing a path negates the integral.

A path has two pieces of lengths 2 and 3. If |f| ≤ 5 on both, what bound does the estimation lemma give?

Let γ(t) be defined for t in [0, 2]. Which reparametrisation is guaranteed to preserve ∫_γ f(z) dz for every continuous f and any piecewise continuously differentiable γ?

• γ(2-s) for s in [0, 2]
• γ(2s) for s in [0, 1]
• γ(s/2) for s in [0, 2]

For γ(t) = 2e^(-it) on [0, 2π], what is ∫_γ conj(z) dz?

• 0
• -8πi
• 8πi

If ∫_γ f(z) dz = 0, then reversing γ gives the same value 0. True or false?

A path consists of a unit circle traversed twice anticlockwise with |f| ≤ 2, plus a segment of length 3 with |f| ≤ 4. Using π < 3.15, what bound does the estimation lemma give?

Let γ1 and γ2 be closed paths from z = 1 back to z = 1 with ∫_γ1 f dz = 3 + i and ∫_γ2 f dz = 2 - i. Since both are loops at the same point, they can be concatenated. A new path goes around γ1 twice, then around γ2 once in reverse. What is the integral?

• 4 + 3i
• 8 + i
• 1 + 2i