now I can prove that the integral is 0 when k< 0
but...for k>0...i still can;t show that it equals to 1...unless assume that ε → 0...
but the question just say that ε > 0
For k > 0
\frac{e^{ikz}}{z-i\epsilon}
pole : z=iε
residule is e^(-kε)
so f(k) = e^(-kε)
for k<0
\frac{1}{(e^{imz})(z-i\epsilon)}
where m=|k|
pole: z=iε
residule: e^(mε)
so f(k) = e^(|k|ε)
umm...how to prove that they equal to 1 or 0?
So...the above result i obtained means that I just consider k>0, right?
then I should find the residule for two case:
k>0 and k<0
then f(k) is equal to the sum of residules
correct?
Complex Variable---definite integral
Show that
f(k) = \frac{1}{2i\pi} \int^{\infty}_{-\infty} \frac{e^{ikx}}{x-i\epsilon}dx =
1, if k>0
0, if k<0
where \epsilon > 0
The attempt at a solution
Consider : \oint \frac{e^{ikz}}{z-i\epsilon}dz
the residule is : e^(-k\epsilon)
so...
Complex Variable---definite integral
Show that
f(k) = \frac{1}{2i\pi} \int^{\infty}_{-\infty} \frac{e^{ikx}}{x-i\epsilon}dx =
1, if k>0
0, if k<0
where \epsilon > 0
The attempt at a solution
1st. step:
Consider : \oint^{\infty}_{-\infty} \frac{e^{ikz}}{z-i\epsilon}dz
the...
A train moves in a straight at a constant speed u. A boy on the train throw a ball of mass m straight ahead, along the motion of the train, with a speed v with respect to himdelf.
a)What is the KE gain of the ball measured by the boy?
b)What is the KE gain of the ball measured by a man...
Let P = linear impulse acted on the stick by the nail
Let I = MI of stick about C
Let Vf = speed of C after impact
Change in linear momentum,
MV - P = MVf......(1)
Change in angular monentum about C,
P(L/4) = wI......(2)
By kinematics,
Vf = (L/4)w......(3)
(1) and (2) give the...
Someone solve this Q without using conservation of energy:
Let P = linear impulse acted on the stick by the nail
Let I = MI of stick about C
Let V1 = speed of C after impact
By conservation of linear momentum,
MV - P = MV1......(1)
By conservation of angular monentum about C,
P(L/4) =...
I'm agree that the rod is not rotating about D, but why Vf = r*w?
r*w is the tangential velocity of the end pt, not the final velocity of the CM
Am I correct?