## Complex Logarithm Wikipedia

### Exponentiation - Wikipedia.

Exponentiation is a mathematical operation, written as b n, involving two numbers, the base b and the exponent or power n, and pronounced as "b raised to the power of n ". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases:.

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### Logarithm - Wikipedia.

In mathematics, the logarithm is the inverse function to exponentiation.That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g. since 1000 = 10 x 10 x 10 = 10 ....

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### Complex conjugate - Wikipedia.

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.That is, (if and are real, then) the complex conjugate of + is equal to . The complex conjugate of is often denoted as ?.. In polar form, the conjugate of is . This can be shown using Euler's formula..

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### Complex logarithm - Wikipedia.

In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers.The term refers to one of the following, which are strongly related: A complex logarithm of a nonzero complex number z, defined to be any complex number w for which e w = z. Such a number w is denoted by log z. If z is given in polar form as z = re i?, where r and ? ....

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### Isomorphism - Wikipedia.

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ???? isos "equal", and u???? morphe "form" or "shape".. The interest in ....

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### Methods of computing square roots - Wikipedia.

Hyperbolic estimates are more computationally complex, because they necessarily require a floating division. A near-optimal hyperbolic approximation to x 2 on the interval [,] is y=190/(10-x)-20. Transposing, the square root is x = -190/(y+20)+10. ... To get the square root, divide the logarithm by 2 and convert the value back. The following ....

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### Argument (complex analysis) - Wikipedia.

In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as in Figure 1. It is a multi-valued function operating on the nonzero complex numbers.To define a single-valued function, the principal value of the ....

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### Binary logarithm - Wikipedia.

In mathematics, the binary logarithm (log 2 n) is the power to which the number 2 must be raised to obtain the value n.That is, for any real number x, = =. For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, and the binary logarithm of 32 is 5.. The binary logarithm is the logarithm to the base 2 and is the inverse function of the ....

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### List of logarithmic identities - Wikipedia.

Complex logarithm identities. The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be ....

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### Hyperbolic functions - Wikipedia.

Hyperbolic tangent. The hyperbolic tangent is the (unique) solution to the differential equation f ? = 1 - f 2, with f (0) = 0.. Useful relations. The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities.In fact, Osborn's rule states that one can convert any trigonometric identity for , , or and into a hyperbolic identity, by expanding ....

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### Cooleyâ€“Tukey FFT algorithm - Wikipedia.

The Cooley-Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size = in terms of N 1 smaller DFTs of sizes N 2, recursively, to reduce the computation time to O(N log N) for highly composite N (smooth numbers). ....

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### P (complexity) - Wikipedia.

In computational complexity theory, P, also known as PTIME or DTIME(n O(1)), is a fundamental complexity class.It contains all decision problems that can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.. Cobham's thesis holds that P is the class of computational problems that are "efficiently solvable" or "tractable"..

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### Laplace operator - Wikipedia.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space.It is usually denoted by the symbols , (where is the nabla operator), or .In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent ....

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### .

Johann Carl Friedrich Gauss (/ g a? s /; German: Gauss [ka?l 'f?i:d?Ic 'ga?s] (); Latin: Carolus Fridericus Gauss; 30 April 1777 - 23 February 1855) was a German mathematician and physicist who made significant contributions to ....

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### Power rule - Wikipedia.

In calculus, the power rule is used to differentiate functions of the form () =, whenever is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule. The power rule underlies the Taylor series as it relates a power series with a function's derivatives.

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### Variety (cybernetics) - Wikipedia.

In cybernetics, the term variety denotes the total number of distinguishable elements of a set, most often the set of states, inputs, or outputs of a finite-state machine or transformation, or the binary logarithm of the same quantity. Variety is used in cybernetics as an information theory that is easily related to deterministic finite automata, and less formally as a conceptual tool for ....

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### Variational Bayesian methods - Wikipedia.

Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random ....

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### Infinite product - Wikipedia.

In mathematics, for a sequence of complex numbers a 1, a 2, a 3, ... the infinite product = = is defined to be the limit of the partial products a 1 a 2...a n as n increases without bound. The product is said to converge when the limit exists and is not zero. Otherwise the product is said to diverge.A limit of zero is treated specially in order to obtain results analogous to those for ....

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### Matrix calculus - Wikipedia.

In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities..

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