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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)
Example 1: Calculate $\log z$ for $z=-1-\sqrt{3}i.$ Solution: If $z=-1-\sqrt{3}i,$ then $r=2$ and $\Theta=-\frac{2\pi}{3}.$ Hence $$\log(-1-\sqrt{3}i)=\ln 2 +i\left(-\frac{2\pi}{3}+2n\pi\right)=\ln 2 +2\left(n-\frac{1}{3}\right)\pi i$$
Complex logarithm functions can be constructed by explicit formulas involving real-valued functions, by integration of /, or by the process of analytic continuation. There is no continuous complex logarithm function defined on all of C ∗ {\displaystyle \mathbb {C} ^{*}} .
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.
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 argz is determined only to a multiple of 2π, each nonzero complex number has an infinite number of logarithms. For example, log(1+i)=ln √ 2+(π/4+2kπ)i =(1/2)ln2+ (8k +1)πi/4.
14 Αυγ 2021 · Logarithmic function. In a similar fashion, the complex logarithm is a complex extension of the usual real natural (i.e., base \(e\)) logarithm. In terms of polar coordinates \(z=re^{i\theta }\), the complex logarithm has the form \(log\,z=log\left ( re^{i\theta } \right )=log\,r+log\, e ^{i\theta } =log\,r+i\theta \).
Trigonometric and hyperbolic functions Complex logarithm Complex power function Definition Properties 1. Complex exponential The exponential of a complex number z = x +iy is defined as exp(z)=exp(x +iy)=exp(x)exp(iy) =exp(x)(cos(y)+i sin(y)). As for real numbers, the exponential function is equal to its derivative, i.e. d dz exp(z)=exp(z). (1)
