What is the MacLaurin formula of higher orders for multivariable functions?

Question

I have an exercise here that is asking me to write the MacLaurin formula of orders II, III, IV for a multivariable function. Example: $ f(x,y)=\cos x \cos y$

Can anyone tell me what the formula looks like for a multivariable function and maybe guide me through this example? Would be much appreciated!

Answer 1

HINT

You can expand separately

• $\cos x=1-\frac{x^2}2+\frac{x^4}{4!}+o(x^4)$
• $\cos y=1-\frac{y^2}2+\frac{y^4}{4!}+o(y^4)$

and then multilply taking the terms to the desidered order.

That is for order IV

$$f(x,y)= \cos x \cdot \cos y =\left(1-\frac{x^2}2+\frac{x^4}{4!}+o(x^4)\right)\left(1-\frac{y^2}2+\frac{y^4}{4!}+o(y^4)\right)=\\=1-\frac{x^2}2-\frac{y^2}2+\frac{x^4}{4!}+\frac{y^4}{4!}-\frac{x^2y^2}4+o(|(x,y)|^4)$$

Note that

• for order II: $f(x,y)= \cos x \cdot \cos y =1-\frac{x^2}2-\frac{y^2}2+o(|(x,y)|^2)$
• for order III: $f(x,y)= \cos x \cdot \cos y =1-\frac{x^2}2-\frac{y^2}2+o(|(x,y)|^3)$

Answer 2

Your example is special in so far as the function is "separated" into two functions of one variable. In order to prove the formula given by Wikipedia consider for fixed $x$ and $y$ the auxiliary function $$g(t):= f(t\,x,t\,y)$$ of one variable $t$, and compute its value at $t=1$, using the Taylor expansion of a function of one variable: $$f(x,y)=g(1)=\sum_{p=0}^n {1\over p!} g^{(p)}(0)+R_n\ ,$$ and compute the higher derivatives $g^{(p)}(0)$ using repeatedly the chain rule. Collecting equal terms you obtain $$g^{(p)}(0)=\sum_{k=0}^p{p\choose k}f_{x^{p-k} y^k}(0,0)\> x^{p-k}\,y^k\ .$$