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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 ...
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].
The dilogarithm along the real axis. In mathematics, the dilogarithm (or Spence's function), denoted as Li 2 (z), is a particular case of the polylogarithm.Two related special functions are referred to as Spence's function, the dilogarithm itself:
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. In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to ...
Other manufacturers used Control-X for this purpose. 10 Control-X was commonly used to cancel a line of input typed in at the terminal. 11 Control-Z has commonly been used on minicomputers, Windows and DOS systems to indicate "end of file" either on a terminal or in a text file. Unix / Linux systems use Control-D to indicate end-of-file at a ...
r = | z | = √ x 2 + y 2 is the magnitude of z and; φ = arg z = atan2(y, x). φ is the argument of z, i.e., the angle between the x axis and the vector z measured counterclockwise in radians, which is defined up to addition of 2π. Many texts write φ = tan −1 y / x instead of φ = atan2(y, x), but the first equation needs ...
More generally, if x = b y, then y is the logarithm of x to base b, written log b x, so log 10 1000 = 3. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b. The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering.
In these cases, the reason may be construction from expanding similar shapes, as is the case for polygonal figures. Logarithmic spiral beaches can form as the result of wave refraction and diffraction by the coast. Half Moon Bay (California) is an example of such a type of beach. [15]