Question: How can I compute this series expansion?

I want to compute the series expansion of i3_r wrt (x, y, z) at point (x=y=z=0):

i2   := (x,y) -> -(1/2)*I*(exp(I*x)*(sin(x)/x)-exp(I*y)*(sin(y)/y))/(x-y):
i3_r := -(1/2)*I*(i2(y,z)-i2(y,x))/(z-x);

My first attempt was to compute this mulltiple series expansion this way:

ordre := 3:
sx := convert( series(i3_r, x, ordre), polynom);
sy := convert( series(sx  , y, ordre), polynom);
sz := convert( series(sy  , z, ordre), polynom);

But this gives me sy=sz=0 whatever the expansion order.

I then do this:

sx :=              convert(series(i3_r , x, ordre), polynom):
sy := add(map(u -> convert(series(u    , y, ordre), polynom), [op(expand(sx))])):
sz := add(map(u -> convert(series(u    , z, ordre), polynom), [op(expand(sy))]));

and obtained non zero results for both sy and sz (but are they are correct ?).

Could you explain me what happens and tell me how to find the series expansion of i3_r wrt (x, y, z) ?


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