Week 4 Worksheet

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Contours

  1. Compute the length of the smooth path γ(t)=3t+it on [3,5].
  2. Let R be the rectangle with vertices 0, 2, 2+3i and 3i. Give formulas for smooth paths γ1, γ2, γ3, γ4 such that the contour Γ=(γ1,γ2,γ3,γ4) traverses the perimeter of R once anti-clockwise.
  3. The reverse of a path γ:[a,b]C is the path γ~:[a,b]C given by γ~(t)=γ(a+bt).
    1. Draw the reverse of γ(t)=it+2(1t) on [0,1].
    2. How would you define the reverse of a contour Γ=(γ1,γ2,γ3)?

Contour Integration

  1. Determine the value of γxy+ix2dz where z=x+iy and γ is each of the following.
    1. The straight line joining 0 to 1+i.
    2. The imaginary axis from 0 to 1.
    3. The line parallel to the real axis from i to 1+i.
  2. Calculate from the definition the following contour integrals.
    1. γ1f where f(z)=1z2 and γ1(t)=2+2eit on [0,2π]
    2. γ2f where f(z)=1(zi)3 and γ2(t)=i+eit on [0,π/2]
  3. Let γ denote the circular path with centre 1 and radius 1, described once anticlockwise and starting at the point 2. Let f(z)=|z|2. Write down a formula for γ and calculate the contour integral of f over γ.

Antiderivatives

  1. For each of the following functions find an antiderivative and calculate the integral along any smooth path from 0 to i.
    1. f(z)=z2sin(z) on C
    2. f(z)=zexp(iz) on C
  2. Put f(z)=|z|2.
    1. Calculate the contour integral of f over the contour that goes vertically from 0 to i then horizontally from i to 1+i.
    2. Calculate the contour integral of f over the contour that goes horizontally from 0 to 1 then vertically from 1 to 1+i.
    3. What does this tell you about possibility of the existence of an antiderivative for f(z)=|z|2?
  3. The \define{reverse} of a path γ:[a,b]C is the path γ~:[a,b]C defined by γ~(t)=γ(a+bt).

    Let γ denote the semi-circular path along a circle with centre 0 and radius 3 that starts at 3 passes through 3i and ends at 3.

    1. Write down a formula for γ.
    2. Let f(z)=1/z2 on the domain C{0}. Calculate the contour integral of f over γ.
    3. Write down a formula for γ~ and calculate the contour integral of f over γ~.
    4. Verify in this case that γ~f=γf.
  4. Let f,g:DC be holomorphic. Let γ be a smooth path in D starting at z0 and ending at z1. Prove that γfg=f(z1)g(z1)f(z0)g(z0)γfg. which is the complex analogue of the integration by parts formula.

The Principal Logarithm

  1. We introduced the principal logarithm via the inversion exp(Log(z))=z for all z0. Here is the other side of that composition.
    1. What is the range {Log(z):z0} of the principal logarithm?
    2. Verify that Log(exp(z))=z for all z in the range of the principal logarithm.
    1. Give an example of complex numbers a,b for which Log(ab)Log(a)+Log(b).
    2. Prove that for any two complex numbers a and b there is nZ such that Log(ab)=Log(a)+Log(b)+2πin holds.
  2. Calculate Log(1+i) and Log(1i).
  3. Given complex numbers a,b with a0 we define ab=exp(bLog(a)).
    1. Calculate ii.
    2. Verify that ab+c=abac for all a,b,c with a0.
    3. Verify that (a1/2)2=a for all a0.
    4. Give an example to prove that (ab)1/2a1/2b1/2 in general.