FUNCTION DIARY
Dr. Rachel Hall
Spring 2006
Due Friday, May 5th. The paper
may be typed (double spaced) or neatly handwritten, or a combination of
both. It should be organized as a paper, rather than a series of
exercises.
Required elements. (not necessarily in
this order)
______ Your function on the real numbers. Describe your function as a function of real
numbers. That is, investigate y=f(x) when x is real.
______ Identification of real and imaginary parts. Find u and v.
______ Mapping. Discuss
the mapping z -> f(z). Draw,
by hand or with technology, a plot or plots that illustrate some important
features of the mapping. Some things you could comment on, depending on
how complicated the mapping is: What does your function do to vertical
lines? Horizontal lines? Rays through the origin? Circles
centered at the origin? All circles and lines? Use color!
______ Domain of definition (and discussion of multiple values, if
appropriate). Find the domain of
definition. Is it open? closed? connected? If you have a
multiple-valued function, how many values does f(z) have?
Indicate how you can make it single valued.
______ Continuity. Is
your function continuous on its domain of definition? If not, where are
the discontinuities? Explain.
______ Behavior at infinity. Find the limit of your function at infinity.
If the limit is undefined, explain why.
Are there points at which lim_(z -> z0) f(z) =
infinity? To answer this question: (1) If your function is
defined on the entire complex plane, you need only check the limit at
infinity. (2) If your function is defined on a punctured plane,
find the limit(s) of f(z) at all the punctures.
______ Modular surface plot. Make a plot of the modular surface of your function.
______ Analyticity. Determine
the singular points of the function and show that your function is analytic
everywhere except at those points. What is the derivative of your
function? the antiderivative?
______ Contour integrals. See HW #9 for the contour integral problems. Cook up a Cauchy Integral Formula problem involving your function and solve it. If your function can be written as f(z) = g(z)/(z-z0)n , you can use the CIF directly; otherwise you’ll have to be a little more creative.
______ Taylor series. Find the Taylor series about z = 0 (if your function is defined there) or about z = 1 (if it’s undefined at 0). Indicate the radius of convergence in the complex plane.
______ Laurent series. If
your function has singular point(s), find its Laurent series (one for each
singular point). If it doesn’t,
find the Laurent series for f(z)/z at the origin. Find the region on which the series
converges.
______ Bibliography Cite
all sources, with page numbers, for theorems you have used without proof (you
need not cite sources for definitions). List the books or web sites you
have used.
Optional elements.
Interesting facts about your function. Does it belong to any interesting classes of functions? To what
other functions is it related? What's its inverse?
Residue Theorem. Think up a nifty Residue Theorem problem
involving your function, and solve it.
Cool stuff. I'm a sucker for
anything neat or unusual!