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$\newcommand{\arg}{\operatorname{arg}}$Say we have the principal branch, $\arg_\tau(z)$. This is defined so that $\arg_\tau(z) \in (-\pi,\pi]$. Why is it necessary to define the limits on the argument of $z$ at all?

Thanks!

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    $\begingroup$ Have you dealt with the complex logarithm function yet? $\endgroup$ – Tim Raczkowski Mar 5 '15 at 4:19
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Holomorphic functions are sometimes defined by a contour integral. The complex logarithm is one such example; more generally one might try to define $$ F(z) = \int_{\mathcal{C}} f(\zeta)~d\zeta, $$ where $\mathcal{C}$ is a smooth curve from a fixed starting point $z_0$ and ending at $z$.

The problem with such a definition is that there are infinitely many such curves $\mathcal{C}$, and in principle the value of $F(z)$ can depend on the precise choice of $\mathcal{C}$. Fortunately, under nice enough conditions (e.g. $f$ is holomorphic) one can show that the value of $F(z)$ is unchanged under a continuous perturbation of the curve $\mathcal{C}$, what is known as a homotopy of curves. This continuous perturbation is subject to some additional constraints, namely that it avoids points where $f$ is not holomorphic.

But now supposing that $f$ has a pole at the origin, this means that admissible curves must avoid the origin. And not all curves in the complex plane minus the origin starting at $z_0$ and ending at $z$ are homotopic; for example, draw the unit circle, then the upper half of the unit circle is not a homotopic curve to the lower half in the punctured complex plane. And this means that $F(z)$ is not well-defined on the complex plane if $f$ has a singularity.

This, for instance, is what happens with the complex logarithm, which is defined by $$ \log z = \int_{\mathcal{C}} \frac{1}{\zeta}~d\zeta, $$ where $\mathcal{C}$ is a curve starting at $1$ and ending at $z$. If one does not specify the branch of the logarithm, then this expression is not well-defined. For if one uses a curve $\mathcal{C}$ that goes counterclockwise around the origin 3 times, and a curve $\mathcal{C}'$ that goes counterclockwise around the origin 0 times, then one finds that $$ \int_{\mathcal{C}} \frac{1}{\zeta}~d\zeta \neq \int_{\mathcal{C}'} \frac{1}{\zeta}~d\zeta. $$ In fact, they will differ by $3(2\pi i)$. One can see this difficulty quickly if one tries to use the rule $\log(xy) = \log(x) + \log(y)$ to define the complex logarithm using the polar expression of a complex number: $$ \log(re^{i\theta}) = \log(r) + \log(e^{i\theta}) = \log(r) + i\theta, $$ but also $$ \log(re^{i\theta}) = \log(re^{i(\theta + 2k\pi)}) = \log(r) + \log(e^{i(\theta + 2k\pi)}) = \log(r) + i\theta + 2k\pi i, $$ so the complex logarithm is only well-defined modulo multiples of $2\pi i$. These multiples correspond to the number of times the curve you use to compute the logarithm goes around the origin.

Once you specify a branch of the logarithm, you essentially prevent curves from going around the origin. Then on this branch, the complex logarithm does not depend on the choice of curve, so it becomes well-defined; and other holomorphic functions defined by contour integrals of functions with a pole at $0$ also become well defined.

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Essentially because $e^{i\theta}$ is a periodic function.

The Euler formula say us that $e^{i\theta}=\cos\theta +i\sin\theta$ so that $e^{i \theta}=e^{i \theta+2k\pi}$ for $k \in \mathbb{N}$.

Thanks to this formula we can write any complex number $z=a+ib$ in a polar form $z=\rho e^{i\theta}$ where $\rho = |z|=\sqrt{a^2+b^2}$ (the positive rooth) is univocally defined, but the argument $\theta$ is not and this cause troubles in defining an inverse function $\log z$ or to extend exponentiation to any complex basis $z^{i\theta}$. So, to have a one to one correspondence we define a principal brunch for $\theta$, usually chosing the interval $(-\pi,\pi]$.

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