Multivariable Taylor series expansion of $\left( \frac{\sin(x)/x}{\sin(y)/y} \right)^2$

Question

I'm trying to find the multivariable Taylor series expansion of

$$ f(x,y) = \left( \frac{\sin(x)/x}{\sin(y)/y} \right)^2 $$

I tried to do this in Matlab (2020b), using the symbolic toolbox

taylor(f,[x y],[0 0],'order',6)

However I get an error saying it can't compute the Taylor expansion. I believe this is due to a divide by zero issue, but I'm not sure.

I can expand the numerator and denominator independently (EDIT: as pointed out, this is not the proper multi variable expansion...it was just to check if $\sin(x)/x$ was causing problems)

$$ \frac{1 - x^2/3 + 2x^4/45}{1 - y^2/3 + 2y^4/45} $$

so I'm not sure why the function expansion in Matlab doesn't work. What am I missing?

EDIT: I should note the above is an example. I'd like to have a general idea how to apply a Taylor series under these circumstances (rational function).

EDIT: The error output is

Error using symengine
Unable to compute a Taylor expansion.

Error in sym/taylor (line 128)
tSym = mupadmex('symobj::taylor',f.s,x.s,a.s,options);
    enter preformatted text here

Answer 1

Update: A better way is to consider the Taylor expansion of $g(t) := f(xt, yt)$ around $t = 0$ and then let $t=1$ in the Taylor expansion.

syms x y t
f = ((sin(x)/x)/(sin(y)/y))^2
f1 = subs(f,x,x*t)
f2 = subs(f1,y,y*t)
g = taylor(f2,t,0,'order',6)
g1 = expand(subs(g,t,1))

By the way, if you use Maple, just use

mtaylor(f,[x=0,y=0],7)

Answer 2

The multivariable expansion is not the quotient of the univariate expansions... $$ f(x,y) = f(0,0) + \frac{\partial f}{\partial x}(0,0) \cdot (x-0)+\frac{\partial f}{\partial y}(0,0) \cdot (y-0)+\frac{1}{2!}\left(\frac{\partial^2 f}{\partial x^2}(0,0) \cdot (x-0)^2+\cdots \right)+\cdots $$

Answer 3

Since $y$ does not depend on $x$, we can make the work by pieces $$\left(\frac{\sin (x)}{x}\right)^2 \implies S_x=1-\frac{x^2}{3}+\frac{2 x^4}{45}+O\left(x^6\right)$$ $$\left(\frac{\sin (y)}{y}\right)^2 \implies S_y=1-\frac{y^2}{3}+\frac{2 y^4}{45}+O\left(y^6\right)$$

$$\frac{S_x}{S_y}=\frac{1-\frac{x^2}{3}+\frac{2 x^4}{45}+O\left(x^6\right) } {1-\frac{y^2}{3}+\frac{2 y^4}{45}+O\left(y^6\right) }$$

Now, make the long division $$\left(1+\frac{y^2}{3}+\frac{y^4}{15}+O\left(y^6\right)\right)+x^2 \left(-\frac{1}{3}-\frac{y^2}{9}-\frac{y^4}{45}+O\left(y^6\right)\right)+x^4 \left(\frac{2}{45}+\frac{2 y^2}{135}+\frac{2 y^4}{675}+O\left(y^6\right)\right)+O\left(x^6\right)$$

Edit

Another solution is to write $$\left(\frac y {\sin (y)}\right)^2 \implies T_y=1+\frac{y^2}{3}+\frac{y^4}{15}+O\left(y^6\right)$$ and expand $S_x\, T_y$ for the same result.