"Can someone confirm that one cannot express a product of two G-functions as a G-function? " Roundabout but: If one has the G-function expression for the Mellin transform of the convolution of two functions then the inverse Mellin operator applied to it would yield the G-function for the product? I am not an expert in sympy or G-functions; just working my way through "The Double Mellin-Barnes Type Integrals and Their Applications to Convolution Theory" and sometimes wandering around in my mind. RayR
On Wednesday, January 27, 2016 at 5:34:26 PM UTC-5, Ondřej Čertík wrote: > > Hi, > > Our Meijer G-function integrator can find integrals (definite and > indefinite) of any function that can be expressed as a Meijer > G-function thanks to the formulas here: > > > http://functions.wolfram.com/HypergeometricFunctions/MeijerG/21/ShowAll.html > > I.e. an integral of a Meijer G-function is also a Meijer G-function. > The definite integral has tons of conditions that SymPy checks, but > the formula for the indefinite integral (i.e. the antiderivative) > always holds. > > Then one converts the final Meijer G-function into elementary > functions if possible, or leaves it as is if it is not possible. This > part is robust. > > What is not robust is how to rewrite a given function into a Meijer > G-function. This is done by the `meijerint._rewrite1` function (btw, > we should expose it as a public function). For example: > > In [1]: meijerint._rewrite1((cos(x)/x), x) > Out[1]: (1, 1/x, [(sqrt(pi), 0, meijerg(((), ()), ((0,), (1/2,)), > x**2/4))], True) > > In [2]: meijerint._rewrite1((sin(x)/x), x) > Out[2]: (1, 1/x, [(sqrt(pi), 0, meijerg(((), ()), ((1/2,), (0,)), > x**2/4))], True) > > In [3]: meijerint._rewrite1((cos(x)/x)**2, x) > > In [4]: meijerint._rewrite1((sin(x)/x)**2, x) > Out[4]: > (1, > x**(-2), > [(sqrt(pi)/2, 0, meijerg(((0,), (1/2, 1/2, 1)), ((0, 1/2), ()), > x**(-2)))], > True) > > In [3] it didn't find the solution, yet a similar expression involving > sin(x) instead of cos(x) works in [4]. > > Let's figure out a systematic algorithm. For that, you start with the > elementary functions, that would be sin(x), cos(x) and "x" in the > above expression, look their G-function representation, and then use > the *, /, +, - and ** operations on the G-functions, that's it. > > Now the problem is, that there doesn't seem to be a formula for a > product of two G functions, e.g. I didn't see one here: > > > http://functions.wolfram.com/HypergeometricFunctions/MeijerG/16/ShowAll.html > > the formula > http://functions.wolfram.com/HypergeometricFunctions/MeijerG/16/02/01/0001/ > that you see there only seems to be using some even more generalized G > function of two variables? It doesn't seem to be useful here. Can > someone confirm that one cannot express a product of two G-functions > as a G-function? > > So a solution is to simply have a robust method to rewrite any > expression as a hypergeometric function and then use the formula here > to convert the hypergeometric function to a G-function: > > > http://functions.wolfram.com/HypergeometricFunctions/HypergeometricPFQ/26/03/01/0001/ > > > There are just a few functions that can be expressed as a G-function > but not as a hypergeometric function, some examples are: Bessel > functions Y, K (for integer order), Whittaker function W, Legendre > function Q_mu_nu and a few others. So for these functions we have to > figure out something else, probably something that we do now. > > Also we can then use the integration formulas here for hypergeometric > functions, so we don't even have to go via G-functions: > > > http://functions.wolfram.com/HypergeometricFunctions/HypergeometricPFQ/21/ShowAll.html > > > It seems the conditions on definite integration are a lot simpler as well. > > > So here is the algorithm for hypergeometric functions, I'll show it on > the (sin(x)/x)**2 example above: > > 1) sin(x) = x * 0F1(3/2; -x^2/4) > > 2) sin(x) / x = 0F1(3/2; -x^2/4) > > 3) (sin(x)/x)**2 = 0F1(3/2; -x^2/4) * 0F1(3/2; -x^2/4) = 2F3(3/2,1; > 3/2,3/2,2; -x^2) > > Where we used the formula for a product of two 0F1 functions: > > > http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric0F1/16/ShowAll.html > > > 4) Finally, rewrite 2F3(3/2,1; 3/2,3/2,2; -x^2) as a G-function, or > integrate directly. > > > A general formula for multiplication of two hypergeometric series is here: > > > http://functions.wolfram.com/HypergeometricFunctions/HypergeometricPFQ/16/ShowAll.html > > > But I can see now that this only expresses the result as a Taylor > series. So maybe I just got lucky that the multiplication of two 0F1 > functions exists (in terms of 2F3), but there is no such formula for a > general PFQ function. Can someone confirm this? > > So then the other idea is that one can identify a hypergeometric > function from the series expansion. Essentially, we expand into a > series, calculate the ratio t_{k+1}/t_k of two successive terms, and > if it is a rational function of "k", then it is a hypergeometric > function and you read the coefficients directly. Let's do the same > example again: > > 1) sin^2(x) = Sum(((-1)**(k - 1)* 2**(2* k - 1)* > x**(2*k))/factorial(2*k), (k, 1, oo)) > > 2) (sin(x)/x)**2 = sin^2(x)/x^2 = Sum(((-1)**(k - 1)* 2**(2* k - 1)* > x**(2*k-2))/factorial(2*k), (k, 1, oo)) > = Sum(((-1)**k * 2**(2*k+1) * x**(2*k))/factorial(2*k+2), (k, 0, oo)) > = Sum((2**(2*k+1) * (-x**2)**k))/factorial(2*k+2), (k, 0, oo)) > > So the series is of the form sum_k t_k*(-x^2)^k where > > t_k = 2**(2*k+1)/factorial(2*k+2) > > 3) Calculate t_{k+1} / t_k: > > In [78]: t_k = 2**(2*k+1)/factorial(2*k+2) > > In [79]: t_k.subs(k, k+1) / t_k > Out[79]: 2**(-2*k - 1)*2**(2*k + 3)*factorial(2*k + 2)/factorial(2*k + 4) > > In [80]: _.simplify() > Out[80]: 2/((k + 2)*(2*k + 3)) > > We write the last formula as: > > 2/((k + 2)*(2*k + 3)) = (k+1) / ((k+3/2)*(k+2)*(k+1)) > > And we read off the hypergeometric function as 1F2(1; 3/2, 2; -x^2). > > One can verify that we got the same answer as before: > > In [64]: hyperexpand(hyper([1], [S(3)/2, 2], -x**2)) > Out[64]: > cos(2⋅x) 1 > - ──────── + ──── > 2 2 > 2⋅x 2⋅x > > In [65]: hyperexpand(hyper([S(3)/2, 1], [S(3)/2, S(3)/2, 2], -x**2)) > Out[65]: > cos(2⋅x) 1 > - ──────── + ──── > 2 2 > 2⋅x 2⋅x > > > And both are actually equal to (sin(x)/x)^2: > > In [75]: (_ - (sin(x)/x)**2).simplify() > Out[75]: 0 > > > One can see, that one should use the exact series expansion, what is > the status of that, I think we had a GSoC on it? > > > It might be, that all the operations that I am doing on the series > have an equivalent operation on the hypergeometric series. > > In conclusion, the logic is simple: either the product or power of > hypergeometric functions exists as a hypergeometric function for the > given set of coefficients (like above) and then there must be an > automated way to determine the final hypergeometric function (in the > worst case by calculating the t_{k+1}/tk ratio like I did above), or > the answer can't be expressed as a hypergeometric function (i.e. when > the t_{k+1}/tk ratio is not a rational function of "k"), and then the > answer is that there is no hypergeometric function that represents the > result, and so we can't integrate it using this method and that's > fine. > > Let me know what you think. > > > This was motivated by https://github.com/sympy/sympy/issues/10464. > > Ondrej > -- You received this message because you are subscribed to the Google Groups "sympy" group. 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