How to expense $(a+b)^\alpha$ into multinomial with $\alpha \in \mathbb{R}$?

Question

As we all know, the binomial expension is as follows $$ (a+b)^2 = a^2 +2ab +b^2. $$ When the power number is a real number, not a integral, how to expense $(a+b)^\alpha$ into multinomial with $\alpha \in \mathbb{R}$?

Answer 1

tetori's reference (Newton's binomial series) is what you want. And note: for a case $(a+b)^\alpha$, say $|a|>|b|$, write $$ (a+b)^\alpha = a^\alpha \left(1+\frac{b}{a}\right)^\alpha $$ and use the series on the second factor. The case $|a|=|b|$ is more complicated. (See "conditions of convergence" on that page.)