WORKSHEET for w1curve.m
-----------------------


WARM-UP
-------

- download into week1 folder:

  w1curve.m  w1curve1.m  w1curve2.m

- matlab test:  type 

  w1curve

  there should be 3 actions.

  1)  a sequence of black dots marking out
  a (discretized) complex contour from z=1
  to z=i

  2)  several blue arrows indicating the
  complex vector of the local derivative

  3)  numerical output of the integral of
  the function f(z) = 3*z.^2 along the 
  plotted complex contour (sb. -1-i)
  

STUDY #1
--------

- action:  same integrand, but different contours
- keyword:  path independence

* design your own complex contour z(t) which
  starts at z(0)=1 and ends at z(1)=i.  use
  any combination of polynomial, trigonometric
  and exponential functions -- you will have
  to differentiate your contour.  CREATIVITY
  COUNTS

- modify a copy of w1curve1.m to utilize your
  new contour

- function check:  "w1curve1(0,0)" should return
  the endpoint.  "w1curve1([0 1],0)" should return
  a row vector of the start/end points!

- quick check:  the blue arrows of the derivative
  z'(t) should be tangent to your contour!

$ what is the numerically approximated (complex) 
  value of the integral of f(z) = 3*z.^2 along this 
  new contour?  compare with other groups in the 
  lab

- change your plot labels

- you should save backup of copies of your most 
  useful modified versions of w1curve1.m


STUDY #2
--------

- action:  special integrand, closed scribbly contour
- keyword:  winding number

* design your own closed "scribble" contour which
  has the form:

  zr = r(cos(2*pi*t), sin(2*pi*t)) .* cos(2*pi*K*t)
  zi = r(cos(2*pi*t), sin(2*pi*t)) .* sin(2*pi*K*t)

  where K is a reasonable integer and r(x,y) is a 
  multivariable polynomial function of your choice
  which is POSITIVE EVERYWHERE on your contour (add
  a constant if you go negative).  warning:  use 
  elementwise arithmetic.  again creativity counts. 
  remember, you will have to differentiate your contour

- modify w1curve.m to utilize this complex contour

- change the integrand function f(z) in w1curve2.m 
  to be:

  f(z) = 1/(2*pi*i)./(z-a)

  by changing the commented (%1) lines.  "a" is a
  complex constant which requires uncommenting
  3 lines of code in w1curve.m and 2 lines in 
  w1curve2.m -- they are marked by a "%1"

- quick check:  start=end, arrows are tangent to 
  the contour & there is a blue star at the origin.
  oddly, the value of the numerical quadrature when 
  a=0 looks like it takes the value of K

- investigate in an orderly fashion what happens 
  for different COMPLEX values of "a" -- watch the
  blue star!

$ what do you believe this integral represents?
  where is the integral expression a continuous
  function of "a"?  why are there discontinuities?

- change your plot labels

- back up your files


STUDY #3
--------

- action:  simple contour, rational function integrand

- start from a fresh set of files

- change the contour to be the unit circle

  zr = cos(2*pi*t)
  zi = sin(2*pi*t)

- the new integrand will be a rational 
  function:

  f(z) = (1/(2*pi*i)) * p'(z)./p(z)

  where p(z) is a 6th-degree polynomial
  whose coefficients are randomly chosen

- the command:

  a = randn(1,6)

  generates a row vector of 6 random real
  numbers and defines a polynomial by 

  p(z) = z.^6 + a(1)*z.^5 + ... a(5)*z + a(6)

  be sure to make "a" a global variable

- the command:

  b = roots([1 a]) 

  gives the 6 complex-valued roots of p(z)=0

- modify w1curve2.m to use the new integrand f(z).
  also the plotting command:

  plot(real(b),imag(b),'b*')

  puts useful stars on your plot.
  
- quick check:  start=end=1, arrows are tangent to
  unit circle of dots & the quadrature seems to give
  an integer less than 7

- change your plot labels

$ draw conclusions