Is it okay to ignore all other branches except principal values? I took only a first course of complex analysis. I have learned some multi-valued functions which consistently extend the classical real functions such as Log and Trigonometric functions. However, I don't see any advantage of considering all the braches. I think it's really convenient to consider $(-\pi,\pi]$ branch so that I can treat those multi-valued function as single-valued functions. Moreover, it's written in wikipedia that if one knows **Riemann Surface** , there is no need to separate branches. Is it okay to just not consider all branches , but to focus on the Principal values?

$\log(z)$ is a smooth multivalued function; it is everywhere differentiable. This is a rather nice feature for doing calculus, complex analysis in particular, because many theorems and calculations only work for continuous or differentiable functions.

For example, $\log(z)$ is an antiderivative of $1/z$ on the whole of the complex plane excluding the origin. $\mathop{\mathrm{Log}}(z)$ is not.

Sometimes, discontinuity is a bigger pain than multivaluedness.

Sometimes you can even have your cake and eat it too: sometimes you work in a domain that intersects the negative real axis, but does not surround the origin. In such a case, you can pick a branch cut for $\log(z)$ in which the logarithm is single-valued and differentiable. You can't do that if you only allow yourself to ever use the principal value.