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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 ...
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 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.
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.
Discrete integral calculus is the study of the definitions, properties, and applications of the Riemann sums. The process of finding the value of a sum is called integration . In technical language, integral calculus studies a certain linear operator .
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.
Sine integral in the complex plane, plotted with a variant of domain coloring. Cosine integral in the complex plane. Note the branch cut along the negative real axis. In mathematics, trigonometric integrals are a family of nonelementary integrals involving trigonometric functions.
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.