I got #6 myself
so only need 1, a/b/c now
1.Evaluate the following integrals
6. let , let and let be continous such that is holomorphic. Show that
for all
(hint: for , apply the cauchy integral formula to ; then let .)
wow this was a lesson in the use of these symbols and latex or w/e
anyways, thats whats left of my assignment due this time tomorrow
if someone could help me get on track I'd appreciate it
#6 looks like the original cauchy statement but i think it's saying that the border may not be holomorphic
Maybe some are unfamiliar with your notation like I was. Ok, let's assume it's what I think it is: means the boundary of a circular disc with center at and radius . Just never seen that notation before. If so then I'll write it as:
(using makes it just unnecessarily cluttered)
So that contour encloses only the pole at the origin and you want to solve it using Cauchy's Integral Formula:
where is analytic in an on the contour. Then:
Here's a check in Mathematica:
Do the other ones just like that but I bet you figured it out already.Code:In[3]:= NIntegrate[(Exp[I*z]/(z^4 - 4*z^2))*2*I* Exp[I*t] /. z -> I + 2*Exp[I*t], {t, 0, 2*Pi}] N[2*Pi*I*D[Exp[I*z]/(z^2 - 4), z] /. z -> 0] Out[3]= 1.5707963267949334 - 8.459899447643693*^-14*I Out[4]= 1.5707963267948966
ok, I kept getting confused at the denominator being 0 but I guess that only matters on the border?
and when I do part b) I get two different solutions depending on what I take as f
example:
for
and
for
is there a reason why I wouldn't use one or the other? or can I not use this approach for some reason
Hey, the denominator is never zero. It's like asking if the denominator is zero for . Never gets ther right? Same dif with the integration path for the first one: . That never gets to the point or . For the second one:
Where the contours on the right side are now only going around each respective pole in accordance with the requirements of Cauchy's Integral Theorem (we can split up that way without problems right? Now:
Also . . . until you get good with these, check them numerically:
If the numerical results don't agree with the symbolic results, and you're pretty sure you've set up the numerical calculations correctly, look for an error in the symbolic results.Code:In[1]:= NIntegrate[(z^3/(2*z^2 - 2))*3*I*Exp[I*t] /. z -> -1 + 3*Exp[I*t], {t, 0, 2*Pi}] Out[1]= -2.220446049250313*^-16 + 3.1415926306916244*I
I screwed up on the original the denominator only one of them is squared
I'll try a few things though to get this
and unfortunately I have no way of checking these numerically since in pure math we don't use mapl or matlab or mathematica or anything so I don't have those