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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...

In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number x to the base b is the exponent to which...

help reach a consensus. › Logarithms and exponentials with the same base cancel each other. This is true because logarithms and exponentials are inverse...

exponents, below), or in terms of the logarithm of the base and the exponential function (§ Powers via logarithms, below). The result is always a positive...

{\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions...

constant of proportionality of this relationship is the natural logarithm of the base b: d d x b x = d d x e x ln ⁡ ( b ) = e x ln ⁡ ( b ) ln ⁡ ( b )...

The imaginary unit or unit imaginary number (i) is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property...

log is often taken by restricting the imaginary part to the interval (−π, π]. This leads to the complex logarithm being a bijective function taking values...

mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation...

super-root of y and the super-logarithm base y of x. The super-root is the inverse operation of tetration with respect to the base: if n y = x {\displaystyle...

by discarding the phase information (contained in the imaginary part of the complex logarithm). It has a focus on periodic effects in the amplitudes...

e {\displaystyle e} is Euler's number, the base of natural logarithms, i {\displaystyle i} is the imaginary unit, which by definition satisfies i 2 = −...

_{b}f(n)=\log _{b}\prod _{n=s}^{t}f(n)\quad } (the logarithm of a product is the sum of the logarithms of the factors) C ∑ n = s t f ( n ) = ∏ n = s t C...

interpreted the quadrature as a logarithm and thus the geometrically defined natural logarithm (or "hyperbolic logarithm") is understood as the area under...

positive imaginary part or a negative imaginary part. This is not always the case; in particular this is not the case for the complex logarithm, the antiderivative...

mathematical notation for logarithms. All instances of log(x) without a subscript base should be interpreted as a natural logarithm, also commonly written...

all imaginary quadratic fields with class number 1. To simplify notation, let L {\displaystyle \mathbb {L} } be the set of logarithms to the base e of...

{x}-1)=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}x^{n}} The natural logarithm (with base e) has Maclaurin series ln ⁡ ( 1 − x ) = − ∑ n = 1 ∞ x n n = − x...

arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields...

}+\sin ^{2}{\theta }\;}}=1} The reason for the use of base e is also now made clear. The imaginary phase constant, i β, can be added directly to the attenuation...