Marcus' function would be well defined if we agreed to use trailing nines wherever the decimal expansion is terminating, this should of course have been specified.
How could that possibly be a bijection? Obviously, z_1=a+ib is mapped to the same point as z_2=a z_1, so it is not an injection.
Marcus has already provided a valid bijection, his "decimal merging" is the classical example of this. Notice how it is also valid in \mathbb{R}^n.
I.e. the multiplicity is the power of the term with the largest negative power in the laurent series of the function?
Does this also mean that an isolated/(essential?) singularity is a pole with infinite multiplicity?
1+x^4 = 1+2x^2+x^4 - 2x^2=(1+x^2)^2-2x^2=(1-\sqrt 2 x + x^2)(1+\sqrt 2 x+x^2).
As to partial fraction decomposition, I suggest you have a look in your textbook or at wikipedia or other webpages.
Why not "guess" a solution of the form x^r, and end up with solutions x^{1/2 \pm i\sqrt{3}/2}, which are essentially the same as Crosson ended up with?
e^{-y^2}=\sum_{n=0}^{\infty}\frac{(-y)^{2n}}{n!}
Is this correct?
I'm inclined to think that:
e^{-y^2}=\sum_{n=0}^{\infty}\frac{(-y^2)^{n}}{n!}
=\sum_{n=0}^{\infty}\frac{(-1)^n y^{2n}}{n!}.?
Kurret: This is not the concept discussed, you have taken the first derivative of some function. The topic is fractional derivatives (for example: what does it mean to take the pi'th derivative of a function?) not the derivatives of functions of fractional powers.,