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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 ...
A branch of is a continuous function defined on a connected open subset of the complex plane such that is a logarithm of for each in . [ 2 ] For example, the principal value defines a branch on the open set where it is continuous, which is the set C − R ≤ 0 {\displaystyle \mathbb {C} -\mathbb {R} _{\leq 0}} obtained by removing ...
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.
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 )} .
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) → ∞.
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.
We can "cut" the plane along the real axis, from −1 to 1, and obtain a sheet on which g(z) is a single-valued function. Alternatively, the cut can run from z = 1 along the positive real axis through the point at infinity, then continue "up" the negative real axis to the other branch point, z = −1.
ln(r) is the standard natural logarithm of the real number r. Arg(z) is the principal value of the arg function; its value is restricted to (−π, π]. It can be computed using Arg(x + iy) = atan2(y, x). Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].