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Letting mathematica compute taylor expansion of implicit function.

by MathematicalPhysicist
Tags: compute, expansion, function, implicit, letting, mathematica, taylor
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May11-12, 01:35 PM
P: 3,217
I have the next function: z^3-2xz+y=0 and I want to find taylor expansion of z(x,y) at the point (1,1,1), obviously I need to define F(x,y,z) as above and use the implicit function theorem to calculate the derivatives of z(x,y), but I want mathematica to compute this to me.

I tried the Series command but I don't know how to use such that it will use the implicit function theorem in the computation.

Any help?

Thanks, I tried looking at the documentation of Mathematica but didn't find anything about taylor expansion of implicit functions.
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May12-12, 01:44 PM
P: 1,666
I think you're going to have to write a routine for that. I can show you how to get the first partial with respect to x. Do the same for y, then iterate:

myFunction = z[x, y]^3 - 2 x z[x, y] + y == 0
myd = D[myFunction, x]
myx1 = First[D[z[x, y], x] /. Solve[myd, D[z[x, y], x]]]

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