Math 455                           Homework                   Saff&Snider
Sec 5.5
_________________________________________________________________________

#7b. The way we did this problem in class is the correct method.  The
     other way you can view this problem is the following, which is
     equivalent to what we did in class.

     a.  The method we used in class.

	 1/(e^z - 1) = 1 / z(1 + 1/2 z + 1/6 z^2 + ...)  = P(z) / z

	 in view of the fact that 1/(e^z-1) has a simple pole at z=0,
	 and where P(z) is analytic at z=0 and P(0) \neq 0.  Hence, it
	 must be the case that

	 P(z) = 1 / (1 + 1/2 z + 1/6 z^2 + ...).

	 Now, if we use long division to divide 1 by the power series
	 in the denomenator, we will obtain the power series representation
	 for P.

     b.  1/(e^z -1 ) = 1 / z(1 + 1/2 z + 1/6 z^2 + ...) = P(z) / z,
	 
	 where P is as described above.  Now, P must have a power series
	 representation like

	 P(z) = a_0 + a_1 z + a_2 z^2 + ...

	 So, we have that

	 1 / (1 + 1/2 z + 1/6 z^2 + ...) = a_0 + a_1 z + a_2 z^2 + ...

	 Hence,

	 (1 + 1/2 z + 1/6 z^2 + ...)(a_0 + a_1 z + a_2 z^2 + ...) = 1.

	 Finally, multiply out the power series on the left and set it
	 equal to the power series on the right (the constant 1).  Then
	 the coefficients a_i's will come out.
_____________________________________________________________________________