Math 3321, Complex Analysis and Integral Transforms
Questions and Answers

Questions and Answers

I often receive math questions from students. Since the answers might be useful to everybody, I am posting them here. I do this anonymously so as not to embarrass the student who asked the question. Ask a question by sending me email at selinger@mathstat.dal.ca (please mention 3321 in the subject line).

Curves and parameterizations

Question (Feb 10): I am confused at what we need to do for the first question #4, finding the parametric representation for 4x^2 + 9y ^2 = 36. Also, I don't know how to find z(t) for the integral questions.

Answer: When you integrate, you have a function you are integrating (f(z), e.g. f(z) = z bar in #18), and you also have a path that you are integrating along.

The path can be given as an implicit equation (e.g. x^2+y^2=1), as a parametric equation (e.g. x=cos t, y=sin t)), as a picture, or in words ("the unit circle", "a square with corners 0, 1, i, 1+i").

No matter how to path is given, you first need to convert it to a parametric form. To do this, you need to know how to convert the most common paths to parametric form.

The most common paths are:

  1. unit circle
    implicit:  x^2 + y^2 = 1
    parametric: x=cos t, y=sin t, where t=0..2pi
    parametric complex: z=cos t + i sin t = e^{it}, where t=0..2pi
  2. circle centered at z0=x0+y0i with radius r
    implicit:  (x-x0)^2 + (y-y0)^2 = r^2
    parametric: x = x0 + r cos t, y = y0 + r sin t, where t=0..2pi
    parametric complex: z = x+iy = z0 + r e^{it}, where t=0..2pi
  3. elliple centered at the origin, with x-intercepts +a,-a, y-intercepts +b,-b:
    implicit:  (x/a)^2 + (y/b)^2 = 1
    parametric: x = a cos t, y = b sin t, where t=0..2pi
  4. curves of the form y=f(x)
    implicit: y=f(x)
    parametric: x=t, y=f(t)
  5. curves of the form x=f(y)
    implicit: x=f(y)
    parametric: x=f(t), y=t
  6. straight line from z0 to z1
    parametric: z=z0 + t(z1-z0), where t=0..1
  7. curves that consist of several pieces, e.g. 4 sides of a rectangle
    Here you need to parametrize each of the pieces individually.

    So for example, in question #4, you are dealing with the equation of an ellipse, therefore 3) above applies. In #20, you are dealing with 4 sides of a square, so 7) applies: you parameterize each of the 4 sides separately. Each is a straight line, so you use 6).

    Path integrals and parametrization of paths are, by the way, also covered in multivariable calculus, so these topics are a repetition in this course. That's why we did not spend so much time on it in class.

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Peter Selinger / Department of Mathematics and Statistics / Dalhousie University
selinger@mathstat.dal.ca / PGP key