$SU_2(\mathbb{C})$ and the characters

Question

i can prove that the irreducible characters $\chi_n$ of $SU_2(\mathbb{C})$ are equal to: $$\chi_n(e^{i\phi})=\frac{\sin((n+1)\phi)}{\sin(\phi)}$$

If i want to give the dimension of the representation belonging to $\chi_n$ i have to compute $\chi_n(1)$, but then $\phi=0$ and thus $\sin(\phi)=0$ thus we have to divide by zero and that can not be the case. The right answer has to be $\chi_n(1)=n+1$ but how to obtain that?

Thanks :)

Answer 1

The function you have is not defined at $\phi = 0$ but the limit as $\phi \to 0$ does exist, and is equal to $n+1$ by L'Hopital's rule.

The manipulations you used to simplify the character into the form you have must have assumed $\phi \ne 0$ but the character is continuous so this justifies being able to evaluate it at $\phi = 0$ by taking the limit.