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The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
Plot of the logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D. In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance.
Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
The integral is reduced to only an integration around a small circle about each pole. application of the Cauchy integral formula or residue theorem Application of these integral formulae gives us a value for the integral around the whole of the contour. division of the contour into a contour along the real part and imaginary part
If f(x) is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure ( quadrature or squaring ...
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.
The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, it is technically not a "principal value". The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function f ( z ) : z = x + i y , {\displaystyle f(z):z=x+i\,y\;,} with x , y ...
A log–log plot of y = x (blue), y = x 2 (green), and y = x 3 (red). Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1. Comparison of linear, concave, and convex functions when plotted using a linear scale (left) or a log scale (right).