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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:
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].
where is the Boltzmann constant (also written as simply ) and equal to 1.380649 × 10 −23 J/K, and is the natural logarithm function (or log base e, as in the image above). In short, the Boltzmann formula shows the relationship between entropy and the number of ways the atoms or molecules of a certain kind of thermodynamic system can be arranged.
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 ...
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.
Given that in general for a closed system with generalized coordinates q i and canonical momenta p i, [3] = =, = =, it is immediate (recalling x 0 = ct, x 1 = x, x 2 = y, x 3 = z and x 0 = −x 0, x 1 = x 1, x 2 = x 2, x 3 = x 3 in the present metric convention) that = = (,) is a covariant four-vector with the three-vector part being the ...
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.
Common values of b are 2, Euler's number e, and 10, and the unit of entropy is shannon (or bit) for b = 2, nat for b = e, and hartley for b = 10. [ 1 ] Mathematically H may also be seen as an average information, taken over the message space, because when a certain message occurs with probability p i , the information quantity −log( p i ...