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Intuitively in complex analysis I know what a Riemann surface is. It is a surface such that at every point on it the value of a function $f(z)$ is single-valued. However, how would I go about finding the equation for such a surface, i.e. if I was to plot one what one I plot?

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    $\begingroup$ THIS might be of some help in understanding the question and providing an answer. $\endgroup$ – Mark Viola Apr 16 '16 at 18:20
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When you are talking about the equation of a surface you have equations like $x^2+y^2+z^2=1$ for a sphere or $z=x^2+y^2$ for a paraboloid in ${\mathbb R}^3$ in mind. But this is not the appropriate way to think about a Riemann surface. A Riemann surface can be any orientable two-dimensional manifold, i.e., "surface" in the intuitive way. In fact the surfaces I mentioned can be the bottom carrier of various Riemann surface structures, whereby the latter will then differ in their local or global conformal structure. To put it bluntly: A Riemann surface is an orientable $2$-manifold on which an angle measurement is defined on all tangent planes.

The surfaces I mentioned inherit an angle measurement in their tangent planes from their imbedding in ${\mathbb R}^3$, whereas the "Riemann surface of the function $z^2=w$" inherits its angle measurement from the angle measurement in the complex $z$-, or $w$-plane (the point $z=w=0$ is special).

To sum it up: A Riemann surface is an orientable $2$-manifold on which you have introduced local coordinates $z_\alpha$ $(\alpha\in I)$ such that in intersections of coordinate patches you have $z_\alpha=\phi_{\alpha\beta}(z_\beta)$ with analytic transition functions $\phi_{\alpha\beta}$.

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$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$As worded the question is a bit awkward to address.

Abstractly, a Riemann surface is a smooth real surface equipped with a holomorphic structure, i.e., a holomorphic curve. Examples include open subsets of $\Cpx$, the complex projective line $\Cpx\Proj^{1}$, tori $\Cpx/\Lambda$ with $\Lambda \subset \Cpx$ a rank-two lattice, and hyperbolic quotients (which tend to be difficult to describe explicitly).

Concretely, certain multi-valued complex functions defined on (open subsets of) the plane, such as logs and roots of entire functions, are associated with Riemann surfaces, and may be understood as subsets of $\Cpx\Proj^{1} \times \Cpx\Proj^{1}$, or as multiple copies of the complex line $\Cpx$ with suitable branch cuts and cross-gluings and added points at infinity. Examples include "the Riemann surface of the $n$th root", $w = \sqrt[n]{z}$ or $w^{n} = z$; "the Riemann surface of $\log$", $w = \log z$ or $\exp w = z$; and "the circle" $w = \sqrt{1 - z^{2}}$, or $z^{2} + w^{2} = 1$.

Alternatively, a homogeneous polynomial $f$ in three variables (whose gradient vanishes only at the origin of $\Cpx^{3}$) defines a Riemann surface embedded in the complex projective plane $\Cpx\Proj^{2}$.

All three classes are difficult to represent geometrically with complete fidelity: Abstract Riemann surfaces are not embedded in anything (and don't have an intrinsic "equation" in the sense you mean), while the concrete "plane curves" naturally sit inside a (real) four-dimensional ambient space, and inevitably undergo distortion (acquire singularities and/or self-intersections) when projected into real three-dimensional space.

That said, a particularly nice example is Costa's surface, a minimal surface in $\Reals^{3}$ of a torus with three punctures given by its Weierstrass embedding.

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You shouldn't expect to be able to look at a surface and say, "oh! here it is: it's all the $z$ that satisfy this equation.

You may not even be able to find an $f(z)$ that globally works. We only know that locally, Riemann surfaces have such a description.

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The standard example (and for me a major motivation for studying Riemann surfaces) is the complex logarithm. The formula usually given for it is really just the branch cut of the entire Riemann surface.

If $z = re^{iθ}$ with $r > 0$ (polar form), then $$w = \ln r + iθ$$ is one logarithm of $z$ [i.e. a branch cut of the Riemann surface]; adding integer multiples of $2πi$ $$w=\ln r + i(\theta + 2n\pi),\ n \in \mathbb{Z}$$ gives all the others [i.e. the entire Riemann surface].

https://en.wikipedia.org/wiki/Complex_logarithm

http://mathworld.wolfram.com/BranchCut.html

The above page actually gives a lot of examples of functions which are not problematic as functions on the real line but which need to be represented either with branch cuts or as Riemann surfaces when considered as functions of complex numbers.

This page has some graphs of Riemann surfaces related to Lambert's W function:

http://mathworld.wolfram.com/RiemannSurface.html

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