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The logarithm has a jump discontinuity of 2 π i when crossing the branch cut. The logarithm can be made continuous by gluing together countably many copies, called sheets, of the complex plane along the branch cut. On each sheet, the value of the log differs from its principal value by a multiple of 2 π i. These surfaces are glued to each ...
A single branch of the complex logarithm. The hue of the color is used to show the argument of the complex logarithm. The brightness of the color is used to show the modulus of the complex logarithm. The real part of log(z) is the natural logarithm of | z |. Its graph is thus obtained by rotating the graph of ln(x) around the z-axis.
Any number log z defined by such criteria has the property that e log z = z. In this manner log function is a multi-valued function (often referred to as a "multifunction" in the context of complex analysis). A branch cut, usually along the negative real axis, can limit the imaginary part so it lies between −π and π. These are the chosen ...
Each value of k determines what is known as a branch (or sheet), a single-valued component of the multiple-valued log function. When the focus is on a single branch, sometimes a branch cut is used; in this case removing the non-positive real numbers from the domain of the function and eliminating π {\displaystyle \pi } as a possible value for ...
Commencing at the point z = 2 on the first sheet we turn halfway around the circle before encountering the cut at z = 0. The cut forces us onto the second sheet, so that when z has traced out one full turn around the branch point z = 1, w has taken just one-half of a full turn, the sign of w has been reversed (because e iπ = −1), and our ...
Using the former definition above, the dilogarithm function is analytic everywhere on the complex plane except at =, where it has a logarithmic branch point. The standard choice of branch cut is along the positive real axis ( 1 , ∞ ) {\displaystyle (1,\infty )} .
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The branch point for the principal branch is at z = − 1 / e , with a branch cut that extends to −∞ along the negative real axis. This branch cut separates the principal branch from the two branches W −1 and W 1. In all branches W k with k ≠ 0, there is a branch point at z = 0 and a branch cut along the entire negative real axis.