Taylor Series in Multiple Variables

[gschran]

Can anyone help me for the leading order terms in the taylor series for the function
f(x,y) = Sqrt(a*x^8+b*x^4*y^4+y^8),
centered at x=0,y = 0 and a,b,c constants?

 

[ross_tang]

Your question really depends on the values of x and y. And your expression doesn't have c. I assumed your function to be:

$$f(x,y)=\sqrt{a x^8+b x^4y^4+c y^8}$$

I only outline the method to obtain a power series here.

$$f(x,y)=c y^8\sqrt{1+\left(\frac{a x^8+b x^4y^4}{c y^8}\right)}$$

By

$$(1+x)^p=\sum _{k=0}^{\infty } \binom{p}{k}x^k$$

,we have:

$$f(x,y)=c y^8\sum _{k=0}^{\infty } \binom{\frac{1}{2}}{k}\left(\frac{a x^8+b x^4 y^4}{c y^8}\right)^k$$

There is another expansion method as well, in which you take out the power x instead of y. It depends on the values of x and y. Since the expansion of $$(1+x)^p$$ is valid for |x| < 1 (|x| = 1 is more complicated).