# Question:How do I find a complicated multi-variable Taylor expansion ?

## Question:How do I find a complicated multi-variable Taylor expansion ?

Hi, and thanks in advance for your help.

My problem starts with an arbitrary function, f, which has two arguments:

f(sqrt(m/(m+1))*(x+1/(2*sqrt(m))), sqrt(m/(m+1))*(y-1/(2*sqrt(m))));

In the problem I'm working on, m is quite large, so both the first and second arguments are more or less equal to x or y, respectively. The step I'm having trouble with is trying to find an expression for the Taylor expansion of this arbitrary function about the point (x,y) itself. My best bet, I think, is to try and use the mtaylor command. I've tried the following:

mtaylor(f(sqrt(m/(m+1))*(x+1/(2*sqrt(m))), sqrt(m/(m+1))*(y-1/(2*sqrt(m)))), [x = x, y = y], 3);

I had hoped that [x=x,y=y] would tell Maple to expand the arbitrary function about the point (x,y), since the help file on the mtaylor function states that the left hand side of both equalities needs to be the variable, and the right hand side the expansion point. I was sure this would not work (given that the left hand sides are not the variables in this case, but rather the more complicated expressions involving the square roots are). I then tried the following:

mtaylor(f(sqrt(m/(m+1))*(x+1/(2*sqrt(m))), sqrt(m/(m+1))*(y-1/(2*sqrt(m)))), [sqrt(m/(m+1))*(x+1/(2*sqrt(m))) = x, sqrt(m/(m+1))*(y-1/(2*sqrt(m))) = y], 3);

This, however, spat back: Error, (in mtaylor) invalid 2nd argument (expansion point). I think my problem is arising because the x,y variables about which I want my expansion are themselves contained in the more complicated arguments given in the first equation.

If I can get this all to work correctly, I'm hoping to arrive at a Taylor expansion of:

f(sqrt(m/(m+1))*(x+1/(2*sqrt(m))), sqrt(m/(m+1))*(y-1/(2*sqrt(m))));

involving sums of terms which themselves contain factors of f, and it's derivatives, exaluated at the point (x,y), with all the m dependence sitting outside of the argument of the function f, contained either as part of the coefficients of the terms or the polynomial factors of the Taylor expansion.

I'm all out of ideas after having tried for a week to figure this out. Any help would be greatly appreciated! Please let me know if you need more clarity on the problem, and thanks again!

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