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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.
A branch of the logarithm is a continuous function L(z) giving a logarithm of z for all z in a connected open set in the complex plane. In particular, a branch of the logarithm exists in the complement of any ray from the origin to infinity: a branch cut. A common choice of branch cut is the negative real axis, although the choice is largely a ...
For complex arguments z with | z | ≥ 1 it can be analytically continued along any path in the complex plane that avoids the branch points 1 and infinity. In practice, most computer implementations of the hypergeometric function adopt a branch cut along the line z ≥ 1. As c → −m, where m is a non-negative integer, one has 2 F 1 (z) → ∞.
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 )} .
However, the important thing to note is that z 1/2 = e (Log z)/2, so z 1/2 has a branch cut. This affects our choice of the contour C. Normally the logarithm branch cut is defined as the negative real axis, however, this makes the calculation of the integral slightly more complicated, so we define it to be the positive real axis.
The sum converges for all complex , and we take the usual value of the complex logarithm having a branch cut along the negative real axis. This formula can be used to compute E 1 ( x ) {\displaystyle E_{1}(x)} with floating point operations for real x {\displaystyle x} between 0 and 2.5.
The polylogarithm function is defined by a power series in z, which is also a Dirichlet series in s: = = = + + +. This definition is valid for arbitrary complex order s and for all complex arguments z with | z | < 1; it can be extended to | z | ≥ 1 by the process of analytic continuation.