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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 ...
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 natural logarithm of x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x. [2] [3] Parentheses are sometimes added for clarity, giving ln(x), log e (x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
Polar form of z = x + iy. Both φ and φ' are arguments of z. All the complex numbers a that solve the equation = are called complex logarithms of z, when z is (considered as) a complex number. A complex number is commonly represented as z = x + iy, where x and y are real numbers and i is an imaginary unit, the square of which is −1.
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 ...
All instances of log(x) without a subscript base should be interpreted as a natural logarithm, also commonly written as ln(x) or log e (x Existence of a prime number between any number and its double In number theory , Bertrand's postulate is the theorem that for any integer n > 3 {\displaystyle n>3} , there exists at least one prime number p ...
The formula as given can be applied more widely; for example if f(z) is a meromorphic function, it makes sense at all complex values of z at which f has neither a zero nor a pole. Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily analysed in terms of the particular case
Analogously, in any group G, powers b k can be defined for all integers k, and the discrete logarithm log b a is an integer k such that b k = a. In number theory, the more commonly used term is index: we can write x = ind r a (mod m) (read "the index of a to the base r modulo m") for r x ≡ a (mod m) if r is a primitive root of m and gcd(a,m) = 1.