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30 Απρ 2024 · The complex logarithm is an extension of the concept of logarithmic functions involving complex numbers (represented by log z). Mathematically, written as. log(z) = log(r ⋅ e iθ) = ln(r) + i(θ + 2nℼ) Here, z = r ⋅ e iθ = the complex number. r = |z| = the absolute value of z. θ = arg(z) = the argument of z and -ℼ < θ ≤ ℼ. ln(r ...
Complex Logarithm. Dive into the fascinating world of engineering maths as you explore the concept of complex logarithm. Grasp the core definition and meaning, broaden your understanding with relevant examples, and discover the fundamental properties of this crucial mathematical element.
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 , defined to be any complex number for which =.
Definition: Complex Log Function. The function \(\text{log} (z)\) is defined as \[\text{log} (z) = \text{log} (|z|) + i \text{arg} (z), \nonumber \] where \(\text{log} (|z|)\) is the usual natural logarithm of a positive real number.
In this topic, we have discussed the logarithm of complex numbers, their definition, showed that log e z is a many-valued f n, discussed the General and principal value of log e z, and showed how it works. Moreover, mentioned some important properties of the logarithm and rules of the logarithm.
14 Αυγ 2021 · In terms of polar coordinates \(z=re^{i\theta }\), the complex logarithm has the form \(log\,z=log\left ( re^{i\theta } \right )=log\,r+log\, e ^{i\theta } =log\,r+i\theta \). We will explore in detail this function in the following section.
Because equation 3.21 yields logarithms of every nonzero complex number, we have defined the complex logarithm function. It is defined for all z 6= 0, and because arg z is determined only to a multiple of 2π, each nonzero complex number has an infinite number of logarithms. For example, √.
