Search results
Results from the WOW.Com Content Network
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 ...
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 ...
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.
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 Government of Canada (French: Gouvernement du Canada) is the body responsible for the federal administration of Canada.The term Government of Canada refers specifically to the executive, which includes ministers of the Crown (together in the Cabinet) and the federal civil service (whom the Cabinet direct); it is alternatively known as His Majesty's Government (French: Gouvernement de Sa ...