itsme

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These are questions asked by itsme

I am trying to solve these equations:

eq1:=cot(theta[n])=(omega[a]^2 - omega[n]^2)/omega[n];

eq2:=cot(theta[n]+ omega[n])=-(omega[b]^2 - omega[n]^2)/omega[n];

for some values of omega[a] and omega[b]. I will in the end create two 3d plots for omega[n] and theta[n] as a funciton of ometga[a] and omega[b]

So for example for some particular values have:

eqs:=subs(omega[a]=0.2, omega[b]=1, [eq1, eq2]);

Then solving,

solve(eqs, [omega[n],theta[n]]);

gives:

I'm however only interested in solutions where omega[n]>=0, but doing this:

solve(eqs, [omega[n],theta[n]], UseAssumptions) assuming omega[n]>=0.0;

returns an empty list. Adding the condition omega[n]>=0.0 to the list of equations also does not work.

Is this a bug or am I missing something? Note, I realize that i can manually go through the entries myself and pick the right solutions, I am just asking how to force maple to do this automagically.

thanks

 

Here is the full code:

restart:

eq1:=cot(theta[n])=(omega[a]^2 - omega[n]^2)/omega[n];

eq2:=cot(theta[n]+ omega[n])=-(omega[b]^2 - omega[n]^2)/omega[n];

eqs:=subs(omega[a]=0.2, omega[b]=1, [eq1, eq2]);

solve(eqs, [omega[n],theta[n]]);

solve(eqs, [omega[n],theta[n]], UseAssumptions) assuming omega[n]>=0.0;

Consider:

> restart:
> int(exp(I*x)/x,x=-infinity..infinity,CauchyPrincipalValue)
                                                                                                   Pi I

> int(exp(I*3*x)/x,x=-infinity..infinity,CauchyPrincipalValue)
                                                                                                   Pi I

> int(exp(I*k*x)/x,x=-infinity..infinity,CauchyPrincipalValue) assuming k::positive;
                                                                                                  2 I Pi

I would expect the last line to also give me Pi I

Mathematica gives a Pi I via:

FullSimplify[Integrate[Exp[I k x]/x, {x, -\[Infinity], \[Infinity]}, PrincipalValue->True, Assumptions->{k>0}],  Assumptions->{k>0}]


Any idea what this could be?

thanks.

Hi:

Say I have an expression like this:

a0:=I*(m*x+mp*x-omega[m]*t-omega[mp]*t);

I can factor out x,t and get an expected result:
collect(a0, [x,t]);

But how would I do it if I want to force Maple to act on the argument of exp?

a1:=exp(a0);

This doesn't work:
collect(a1, [x,t])

This will do what I want for this expression but maybe not in general:

Hi:

I would like to ask maple to "recognize" a Fourier transform in a (possibly complicated) expression.

Maple defines a Fourier transform in this way:
convert(fourier(f(x), x, k), Int);

So if f(x)=1 we should get something that is proportional to Dirac(k), which we do:
fourier(1, x, k);

...but given an expression
exp1:=Int(exp(-I*x*k), x = -infinity .. infinity);

... how would tell...

I am trying to see if Maple can help me find a closed form for an infinite series. I do this:

 

restart:

s:=Sum(n^2/((Pi^2*n^2+beta^2*epsilon^2)*(exp(n))^4), n = 1 .. infinity);

value(s);

 

Maple returns with:

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