How can I find Cauchy principal value. of this integral
\[ n(x) = \int_{a}^{b} \frac{d \omega}{\omega ' ^2 - x^2} \] Where $ a<x<b $
I case $a = 0, b = 3, x = 1$ We get
\[ n(1) = \int_{0}^{3} \frac{d \omega}{\omega ' ^2 - 1^2} = −0.3465735902799727 \] The result shown is the Cauchy...
\[ \int_{0}^{\inf} \frac{e^{-\frac{(x-a)^2}{b}}}{x^2-c^2} dx\] or \[ \int_{0}^{constant} \frac{e^{-\frac{(x-a)^2}{b}}}{x^2-c^2} dx\]
maybe application Residue theorem integral ? because this problem same the kramers kronig relation?
$\dot{r} $ mean full derivative of r by dt because \[ r=r(q_1,...,q_n,t) \] and \[ q_n = q_n(t) \]
any $q_n$ as function of time so $\dot{r}$is formed by taking the derivative with respect to dt for $( q_1,...,q_n,t )$
from problem I find \[ r = r_0 + At \] \[ x_0 = 3 + 2t\] \[ y_0 = -1 - 2t\] \[ z_0 = 1 + t\] and \[ A = (2,-2,1)\]
but i don't understand What is the distance of closest approach?
someone tell me to a formula please.
I select \[ y_1= c_1x^r\] and \[ y_2= c_2x^s\]
so \[ (x+1)x^2y_1'' + xy_1' +(x+1)^3y_1 = 0\] and
\[ (x+1)x^2y_2'' + xy_2' +(x+1)^3y_2 = 0\] i find first and second deriative of y1 and y2
I get two equations
\[r(r-1)(x+1) +rx - (x+1)^3 = 0 \] \[s(s-1)(x+1) +sx - (x+1)^3 = 0 \]
in problem b from \[ y_1y_2 = c \] so I was able to specify that \[ y_1 = c_1x^2 \] abd \[ y_2 = c_2x^{-2} \]
Correspond to \[ y_1y_2 = c_1c_2 = c = constant \] then I can find \[ y_1', y_1'', y_2',y_2'' \]
So. I can solve \[2p_1p_2 +p_2' = 0\]
But in problem C, I have no idea, so I assign \[...
I don't understand, please ckeck
Let V=\Bbb{R}^2 and {u=(u_1,u_2), v=(v_1.v_2)}\in\Bbb{R}^2 , {k}\in \Bbb{R} define of operation u\oplus v = (u_1+v_1,u_2+v_2) and k \odot u =(2ku_1,2ku_2) check V is vector over field \Bbb{R} ?
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