"How do you mean: "if x is invertible"?"
x is an element of a general algebra, not merely a real or complex number but a multivector or matrix or similar such object that can be raised to integer powers and summed and so exponentiated. I've encountered this in quantum mechanics ..
"Reduced Exponential"
I am interested in what I call the "reduced exponential"
Sum_i=1 to infinity x^(i-1) / i!
where x is a general element in an algebra of interest.
Only when x is invertible is the reduced exponential equivalent to (exp(x)-1) /x .
Obviously we have a "reduced...
"Why shouldn't it have precisely the correct momentum?"
Because its just not credable. This is angular momentum so the photon has to have /exactly/ the right trajectory. The absence of consideration of orbital kinetic energy alarms me too.
But if the electron is absorbing a photon to gain energy, why should this photon have precisely the correct mementum too? If it sheds surplus momentum by emitting a photon, it will lose energy.
"Of course"? You mean that's the theory? An incoming photon already of precisely absorbable wavelength just happens to have precisely the right trajectory to provide the necessary angular momentum too?? What are the odds?
If the atom was scattering photons it might be more credable but isn't...
In the Bohr atomic model, electrons move between shells having angular momentum L_n = n h_bar where n is the shell number and the difference in shell energies E_n2-E_n1 matches the Rydberg energy of the emitted or absorbed photon.
My question is: what hapened to the angular momentum...
Obviously if you can do the indefinite integral the definite is easy but I suspect the definite integral (corresponding to the potential of a full ring) is likely to be easier. To do it with contour you'd obviously have to close the contour from q1 to q2 somehow.
I don't have Mathematica...
Thanks for that. I've not tried it. I'm not that keen on tackling this analytically since my calculus is very rusty and I was hoping it would be a standard problem, already tackled.
The definite integral between the quadratic roots of f(x) / sqrt(Q(x)) for f(x) ammoung the basic fundamental...
Recently I fruitlessly asked if anybody could help with the definite integral of
sin(a x) / ( x sqrt(-(x-q1)(x-q2)) )
from x=q1 to x=q2 where a,x, q1 and q2 are all real.
If the sqrt wasn't there one could use contour integration and consider residues at q1,q2 and 0 but with the sqrt I am...
Worth pointing out here that since in the standard EPR experiment the observations are considered as spacially separated rather than timelike sepertied events, it is meaningless to speak of Alice making her measurement "before" Bob makes his as there will be some observers who percieve Bob's...
Can you elaborate?
Going complex, we might consider the imaginary part of
exp ( i a z) / ( z sqrt(-(z-q1)(z-q2) )
or more generally f(z) / sqrt(-(z-q1)(z-q2))
as z moves along the real axis from q1 to q2 , the two real roots of quadratic Q(x)=x^2+bx+c.
I am interested in the definite integration of
f(x) / sqrt(-Q(x))
over the range of x for which quadratic
Q(x) = x^2 + b x + c is negative .
f(x) = sin(a x)/x in particular but others too.
Can anybody point me at any known formulae?
TIA.
Oh well in that case, in the absence of provenance for the Office-Shredder claim, I dub the unique solution to x tan(x)=y in
[(k-half)pi,(k+half)pi] for nonzero integer k to be the k-th Bellian function of y.
Written capital Beta sub k (y) to distinguish from the Bessel and Bell and , er, Beta...
Does the function f(x) = x tan(x) have a name? I am particularly interested in the solutions to x tan(x) = k for integer k. Do these numbers have an accepted name or notation?
TIA.