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30 Απρ 2024 · Complex Logarithm. 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.
§ Complex Logarithm Function. The real logarithm function ln x is defined as the inverse of the exponential function — y = ln x is the unique solution of the equation x = ey. This works because ex is a one-to-one function; if x1 6=x2, then ex1 6=ex2.
Definition. The complex logarithm for z 2C is log(z) := fw 2C : exp(w)=zg. Theorem (1.7.3). Let z;w 2 , then log(z)=lnjzj+iarg(z); log(zw)=log(z)+log(w); and log(1=z)= log(z): Definition. The principle logarithm is Log(z):=lnjzj+iArg(z). Definition. A branch cut is L z 0;q:= fz 2C : z = z 0 +reiq;r 0g, giving the cut plane D 0;p:= CnL 0;p.
Chapter 13: Complex Numbers. Sections 13.5, 13.6 & 13.7. 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.
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} ^{*}} .
In these notes, we examine the logarithm, exponential and power functions, where the arguments∗ of these functions can be complex numbers. In particular, we are interested in how their properties differ from the properties of the corresponding real-valued functions.†. 1. Review of the properties of the argument of a complex number.
Trigonometric and hyperbolic functions Complex logarithm Complex power function De–nition Properties 1. Complex exponential The exponential of a complex number z = x +iy is de–ned 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 ...
