Hia all.
I would like to request comments and maybe clarificaions on the task and solutions provided in the attached PDF.
It was one of our homework tasks and, IMO the provided solution does not solve the task at hand.
I'm currently in a dispute with the chief TA about this and so far...
Sorry.
The way hinted to me in the paper is that we have to take that t^(-1/2) definition and use fubini's thm.
This leads to the aforementioned
\int_{0}^{\infty}\frac{u^{2}}{1+u^{4}}du
I am perfectly able to integrate the RHS of the equation you've posted, but...
That is the answer I, too, have derived from the solutions Mathematica gave me.
What I need to know is which method to use to get from the left side to the right side.
That is a step non-obvious to me and it has to be.
Homework Statement
The initial problem is to calculate
\int_{-\infty}^{\infty}\cos(x^{2})dx using
t=x^{2}
and then
t^{-\frac{1}{2}}=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-tu^{2}}du
Homework Equations
The Attempt at a Solution
I have, by transformation and use of the...
Homework Statement
Prove that J_{n}, Y_{n} satisfy
x^{2}*y''(x)+x*y'(x)+(x^{2}-n^{2})*y(x)=0
where n\inZ and x\in(R_{>0}
Homework Equations
The standard definitions of the bessel integrals as given here:
http://en.wikipedia.org/wiki/Bessel_Functions
The Attempt at a Solution...
What you are looking for seems to be the variance of measurements.
For the std-deviation of the mean value:
s_{m}=\frac{s}{\sqrt{n}}=\sqrt{\frac{1}{n(n-1)}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}
here then holds for n->infinity
s_{m}\rightarrow0\; s\rightarrow\sigma
with...
Why though? The way known to me to diagonalize a symmetric matrix is TAT^-1 = D
with orthogonal matrix T.
Following is the way I think I've solved it.
Taking off from hunt's equation...
Yep.
I've got that hint by a friend yesterday night which led me to:
\int e^{-\frac{1}{2}<x,T^{-1}TAT^{-1}Tx>-i<k,T^{-1}Tx>}\Rightarrow\int e^{-\frac{1}{2}<Tx,(TAT^{-1}=D)Tx>-i<Tk,Tx>}\Rightarrow\int e^{-\frac{1}{2}<y,Dy>-i<k',y>}d^{n}y
by using that TAT^-1 = D , with D-diagonal. defining...
Homework Statement
Let A be a real, symmetric positively definite nxn - matrix.
f:\mathbb{R}^{n}\rightarrow\mathbb{R}\; s.t\;\vec{x}\rightarrow e^{-\frac{1}{2}<\vec{x},A\vec{x}>}
Show that the FT of f is given by:
\hat{f}(\vec{k})=\frac{1}{\sqrt{\det...