a). integral over gamma of e^(iz) / z^2 dz, where gamma(t) = e^(it), 0=<t=<2 pi
( e here is exponential function).
Please use basic definitions and power series representation of analytic functions to do so.

b). integral over gamma of sin(z)/z^3 dz ( same gamma and values of t as above)

Solution Summary

This shows how to evaluate to integrals over gamma (power series representations)

By considering appropriate series expansions, prove that the powerseries expansion of the product of the (infinitely many) exponential factors e^{(x^i)/i}, i = 1, 2, 3, ..., is 1 + x + x^2 ... for |x| <1.
By expanding each individual exponential factor in the product and multiplying out, also show that the coefficient of x^1

Hello,
I am in a fast-paced Calculus course where I must learn new concepts each week; I find it challenging to grasp the concepts while remaining on-pace and I am experiencing great difficulty. I have a few weeks before my semester is over, and thankfully, I have a passing grade.
I am really finding it difficult to grasp

Consider the differnetial equation
y'(x) + xy(x) = 0 with y(0) = 0
Look for a solution of this problem of the form
y(x) = A + B + Ce^-x + De^-1/2x^2
Use the fact that y must satisfy the equation and the initial conditions to identify the constants A,B,C and D. By setting u = -x^2/2 in the powerseries for f(u) = exp{u},

Define the set R[[X]] of formal powerseries in the indeterminate X with coefficients from R to be all formal infinite sums
sum(a_nX^n)=a_0 +a_1X+a_2X^2+...
Define addition and multiplication of powerseries in the same way as for powerseries with real or complex coeficients,i.e extend polynomial addition and multiplication t

FIND POWERSERIES SOLUTIONS IN POWERS OF X OF THE D.E.
y''-xy'+(3x-2)y=0
and
FIND POWERSERIES SOLUTION OF EA. OF THE INITIAL-VALUE.
y''+xy'-2y=0, y(0)=0, y'(0)=1
ON BOTH OF THE PROBLEMS I GOT STUCK ON THE RECURRENCE FORMULA AND GETTING POWERS OF X.

An incandescent lightbulb rated at 100 W will dissipate 100 W as heat and light when connected across a 110-V ideal voltage source. If three of these bulbs are connected in series across the same source, determine the power each bulb will dissipate.