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A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued function on the plane minus that curve. Branch cuts are usually, but not always, taken between pairs of branch points.
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 principal values. This is the principal branch of the log function.
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 ...
This is commonly done by introducing a branch cut; in this case the "cut" might extend from the point z = 0 along the positive real axis to the point at infinity, so that the argument of the variable z in the cut plane is restricted to the range 0 ≤ arg(z) < 2π. We can now give a complete description of w = z 1/2.
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 a region , where ¯ is the complex conjugate of and (, ¯) and (, ¯) are functions of and ¯. [ 11 ] Generalized analytic functions have applications in differential geometry , in solving certain type of multidimensional nonlinear partial differential equations and multidimensional inverse scattering .
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 as a possible value for Arg z.
The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i The graph of y = W(x) for real x < 6 and y > −4. The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1. The minimum value of x is ...