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These are answers submitted by digerdiga

Does noone know ? :-(

@adri: thx for your help. for the second part what exactly does the x-> mean?as far as I know it is just an asignment like f: x-> f(x) so whats the difference?

why cant I just omit the x-> ??


@ axel: what do you mean? sin(x) square wave? what?

edit: @ axel ah yeah^^thats what i mean

weird now it works...

@Carl Love 

1. Instead of making a series approximation to the Wronskian, how about first testing via plots that your approximation to the HeunG functions is correct.

Which approximations do you mean in particular?Plotting the series approximations of the heung functions?I've done that... they were correct...

2. Oddly enough when calculating the function HeunG(c,q,a,b,g,d,t) as a series expansion with the coefficients from above (with fixed p and then plotting the function) I can go up to 400 with no problem... It's just the combination of the sum within a sum in this wronskianseries somehow which limits the number of terms to 20/30...I dont know why is that...

3. I dont get what you mean...I'm interested in the 2 point connection problem thus the wronskian here gives me the information about the coefficients. i.e. I want to write the solution about t=0 as a linear combination of the solutions about t=1... thus the mixing...

4. the t is evaluated at t=1/2...since the wronskian in my case is independent of t anyway (because the entire solution is a solution of an ode with no first derivative i.e. of schroedinger type)

5. See 1. what approximations do you mean? same as in 1?Didnt do it since the Heung approximations itsself were right... So should then be the derivatives


All in all I have solved the problem for s=1/2 (though only in mathematica yet but the code as above should be right) but unfortunately when decreasing s there is some limiting value around s~0.15 (I think its where the fourth bound state comes in) where either the convergence gets really bad or vanishes at all...

Like functions as the riemann zeta function which is originally only valid for values larger than 1, one can analytically continue those functions into the entire z plane excluding the singularity which arises from the classic series expansion...Given that... how exactly is that problem solved in maple?i.e. When Calculating the wronskian with the implemented heun functions I can go down to say s=0.1 which is smaller then the limit value s~0.15... But the other way by expanding the wronskian into a series doesnt work for these particular values... only for s>~0.15...

doesnt it matter where the package declarations stand?

copy paste doesnt work for me. it just copies the entries of maple like a picture...

you would not be able to copy it...therefore I inserted a maple spreadsheet...

I use maple16 and Linux Debian

what run do u mean?

plotting deltag takes about 10sec

plotting diffdeltag took about a few Minutes

Integrating till 20 about a few minutes aswell...


i havent changed my notation.

if you are refering to the -ln(2)*k term i have eliminated this term by the factor exp(-I*ln(2)*sqrt(x-4)) in the wronskian because the argument of yg would otherwise be discontinuous..dont worry about it.

added a .mw file

yg := Determinant(subs(c = (1+s)/(1-s), q = 2*k^2*s/(1-s)+k*(1+9*s)/(2*(1-s))+k^2+2*k, a = k, b = k+1/2, g = 1/2, d = 2*k+3, s = 1/2, k = -1+I*sqrt((1/4)*x-1), t = 1/2, sqrt(x-4) = k, Wronskian([(1-t)^(I*sqrt((1/4)*x-1))*HeunG(c, q, a, b, g, d, t), (1-t)^(I*sqrt((1/4)*x-1))*exp(-I*ln(2)*sqrt(x-4))*HeunG(1-c, -q+a*b, a, b, d, g, 1-t)], t)))

-((1/2)^(((1/2)*I)*k))^2*HeunG(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)*exp(-I*ln(2)*k)*HeunGPrime(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)-((1/2)^(((1/2)*I)*k))^2*exp(-I*ln(2)*k)*HeunG(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)*HeunGPrime(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)


deltag := argument(yg)

argument(-((1/2)^(((1/2)*I)*k))^2*HeunG(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)*exp(-I*ln(2)*k)*HeunGPrime(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)-((1/2)^(((1/2)*I)*k))^2*exp(-I*ln(2)*k)*HeunG(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)*HeunGPrime(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2))


plot(deltag, k = -35 .. 35)


diffdeltag := Im(diff(ln(yg), k))

Im(((2*I)*((1/2)^(((1/2)*I)*k))^2*HeunG(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)*ln(2)*exp(-I*ln(2)*k)*HeunGPrime(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)-((1/2)^(((1/2)*I)*k))^2*(((3/4)*I-(3/2)*k)*(D[2](HeunG))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+((1/2)*I)*(D[3](HeunG))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+((1/2)*I)*(D[4](HeunG))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+I*(D[6](HeunG))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2))*exp(-I*ln(2)*k)*HeunGPrime(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)-((1/2)^(((1/2)*I)*k))^2*HeunG(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)*exp(-I*ln(2)*k)*((-(3/2)*I+k)*(D[2](HeunGPrime))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+((1/2)*I)*(D[3](HeunGPrime))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+((1/2)*I)*(D[4](HeunGPrime))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+I*(D[5](HeunGPrime))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2))+(2*I)*((1/2)^(((1/2)*I)*k))^2*ln(2)*exp(-I*ln(2)*k)*HeunG(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)*HeunGPrime(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)-((1/2)^(((1/2)*I)*k))^2*exp(-I*ln(2)*k)*((-(3/2)*I+k)*(D[2](HeunG))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+((1/2)*I)*(D[3](HeunG))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+((1/2)*I)*(D[4](HeunG))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+I*(D[5](HeunG))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2))*HeunGPrime(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)-((1/2)^(((1/2)*I)*k))^2*exp(-I*ln(2)*k)*HeunG(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)*(((3/4)*I-(3/2)*k)*(D[2](HeunGPrime))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+((1/2)*I)*(D[3](HeunGPrime))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+((1/2)*I)*(D[4](HeunGPrime))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+I*(D[6](HeunGPrime))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)))/(-((1/2)^(((1/2)*I)*k))^2*HeunG(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)*exp(-I*ln(2)*k)*HeunGPrime(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)-((1/2)^(((1/2)*I)*k))^2*exp(-I*ln(2)*k)*HeunG(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)*HeunGPrime(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)))


plot(diffdeltag, k = -25 .. 25)




I wish to calculate the integral



int(VectorCalculus:-`*`(diffdeltag, ln(VectorCalculus:-`+`(k^2, 4))), k = 0 .. infinity)

Warning,  computation interrupted




or maybe more maple conform after partial integrating and resumming (delta+Pi/2) in the first and second term in order to be finite


int(VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`+`(deltag, VectorCalculus:-`*`(Pi, 1/2)), 2), k), 1/VectorCalculus:-`+`(k^2, 4)), k = 0 .. infinity)

Warning,  computation interrupted






negelecting the first term of partial integration

















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Download mapleprimes.mwmapleprimes.mw

first of all thank you for all your answers... i dont quite get why maple cant differentiate an argument (same as with abs apparently) but here is how i solved it:

If i call yg(k) the function which contains the HeunG functions i know it has some complex value say R(k)*exp(I*delta(k))

So diff(yg(k),k)=yg(k)*(diff(R(k),k)/R(k)+I*diff(delta(k),k))



Which plots the right result.I can show you what i meant if im back at uni. Unfortunately for values k larger than 25/30 the function doesnt have the right behaviour anymore. I guess it is due to inaccuracies. Still very unfortunate.

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