= Monodromy
{wiki}
Monodromy records how a <germ of a holomorphic function> changes after <analytic continuation> around a closed path. The source point returns to itself, but the continued <germ of a holomorphic function> need not. For $z(t)=e^{it}$, $0\le t\le2\pi$, starting with <holomorphic square root> value $1$ gives $w(t)=e^{it/2}$ and hence $w(2\pi)=-1$; starting with <complex logarithm> value $0$ gives $L(t)=it$ and hence $L(2\pi)=2\pi i$. After $m$ counterclockwise turns the values are $(-1)^m$ and $2\pi im$. Thus <monodromy> exchanges the two square-root <branches of a multivalued function>, whereas it shifts through infinitely many logarithm <branches of a multivalued function>. <Branch Points and Branch Cuts by Rodolfo R. Rosales>[Rosales, Branch Points and Branch Cuts, section 1 and Example 2.4]{ref}
\Image[/logarithm-square-root-monodromy.png]
{title=One closed loop in the source plane lifts to an open semicircle for the square root and an open vertical segment for the logarithm}
The three panels use the same parameter $t$. They show <analytic continuation> through local <holomorphic> patches, including across a chosen <branch cut>; repeatedly evaluating the <principal logarithm> would instead reset its <imaginary part> at the cut. <Monodromy> makes the obstruction to a global <branch of a multivalued function> visible and motivates the use of <Riemann surfaces> as domains on which continued values stay distinct.
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