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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.
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 matter of convenience. The logarithm has a jump discontinuity of 2 π i when crossing the branch cut. The logarithm can be made ...
For example, the principal branch has a branch cut along the negative real axis. If the function L ( z ) {\displaystyle \operatorname {L} (z)} is extended to be defined at a point of the branch cut, it will necessarily be discontinuous there; at best it will be continuous "on one side", like Log z {\displaystyle \operatorname {Log} z ...
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 principal values. This is the principal branch of the log function. Often it is defined using a capital letter, Log z.
Branch and cut [1] is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values. [2] Branch and cut involves running a branch and bound algorithm and using cutting planes to tighten
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) → ∞.
The integral defines a solution in the right half-plane Re z > 0. ... In the second formula the function's second branch cut can be chosen by multiplying with (−1) p.