goodphy
- 212
- 8
Hello.
Let's have any non-zero complex number z = reiθ (r > 0) and natural log ln applies to z.
ln(z) = ln(r) + iθ. In fact, there is an infinite number of values of θ satistying z = reiθ such as θ = Θ + 2πn where n is any integer and Θ is the value of θ satisfying z = reiθ in a domain of -π < Θ < π. So, ln(z) can be written as ln(z) = ln(r) + i(Θ + 2πn).
ln(z) is multiple-valued function as each Θ + 2πn with different n results in different ln(z), although Θ + 2πn are essentially same angle between radial line from the origin to z and the positive real axis in the complex plane (in other word, Θ + 2πn indicates same z).
In order to make single-valued and continuous function, the domain of θ needs to be restricted somehow. The textbook of complex analysis says a way to restrict domain of the multiple-valued function ln(z) to make the single-valued function F is lik this; F = ln(z) = ln(r) + iθ (α < θ < α + 2π).
My question is why the domain is restricted by (α < θ < α + 2π), instead of (α ≤ θ < α + 2π)? I think later domain also can be used to F. F is continuous from θ = α itself to the point infinitesimally close to α + 2π and is single-valued over this domain.
I think tihs is important question for me as I'm trying to understand the concept of the branch cut and branch point for the integral on a contour in the complex plane.
Let's have any non-zero complex number z = reiθ (r > 0) and natural log ln applies to z.
ln(z) = ln(r) + iθ. In fact, there is an infinite number of values of θ satistying z = reiθ such as θ = Θ + 2πn where n is any integer and Θ is the value of θ satisfying z = reiθ in a domain of -π < Θ < π. So, ln(z) can be written as ln(z) = ln(r) + i(Θ + 2πn).
ln(z) is multiple-valued function as each Θ + 2πn with different n results in different ln(z), although Θ + 2πn are essentially same angle between radial line from the origin to z and the positive real axis in the complex plane (in other word, Θ + 2πn indicates same z).
In order to make single-valued and continuous function, the domain of θ needs to be restricted somehow. The textbook of complex analysis says a way to restrict domain of the multiple-valued function ln(z) to make the single-valued function F is lik this; F = ln(z) = ln(r) + iθ (α < θ < α + 2π).
My question is why the domain is restricted by (α < θ < α + 2π), instead of (α ≤ θ < α + 2π)? I think later domain also can be used to F. F is continuous from θ = α itself to the point infinitesimally close to α + 2π and is single-valued over this domain.
I think tihs is important question for me as I'm trying to understand the concept of the branch cut and branch point for the integral on a contour in the complex plane.