Domain of single-valued logarithm of complex number z

In summary, the conversation discusses the multiple-valued function ln(z) and the need to restrict its domain in order to make it a single-valued and continuous function. The textbook suggests using the range (α < θ < α + 2π) to restrict the domain, but the individual has a question about why the range (α ≤ θ < α + 2π) cannot also be used. They eventually come to the conclusion that one of the end points of the range of θ can be included. There is also mention of a potential misprint in the textbook.
  • #1
goodphy
216
8
Hello.

Let's have any non-zero complex number z = re (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 = re such as θ = Θ + 2πn where n is any integer and Θ is the value of θ satisfying z = re 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.
 
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  • #2
You are correct. You can include one of the end points of the range of θ
 
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Likes goodphy
  • #3
FactChecker said:
You are correct. You can include one of the end points of the range of θ

I see. I was worrying whether the textbook was misprinted. Other mathematical physics book says the domain of the single-valued function is F(z) = ln(z) = ln(r) + iθ is α ≤ θ < α + 2π.

I think my book was misprinted:)
 

Related to Domain of single-valued logarithm of complex number z

1. What is the domain of the single-valued logarithm of a complex number?

The domain of the single-valued logarithm of a complex number z is the set of all complex numbers excluding 0. This means that any complex number z can be used as an input for the single-valued logarithm as long as it is not equal to 0.

2. Why is 0 excluded from the domain of the single-valued logarithm of a complex number?

This is because the single-valued logarithm function is not defined at 0. This can be seen intuitively by considering the inverse relationship between exponentiation and logarithms. The value of a logarithm is the exponent that the base must be raised to in order to equal the input. Since 0 cannot be raised to any power to equal a non-zero number, it is not included in the domain of the logarithm function.

3. Can the single-valued logarithm be extended to include 0 in its domain?

Yes, the single-valued logarithm can be extended to include 0 in its domain by using a branch cut. A branch cut is a mathematical tool that allows for the extension of a function's domain beyond its traditional limits. In this case, the branch cut would be placed along the negative real axis, allowing for the inclusion of 0 in the domain of the single-valued logarithm function.

4. How is the single-valued logarithm of a complex number calculated?

The single-valued logarithm of a complex number is calculated using the following formula: log(z) = ln|z| + iθ where z = |z|e is the polar representation of the complex number and ln|z| is the natural logarithm of the magnitude of z. This formula can be derived from the definition of the complex logarithm and properties of exponential functions.

5. What is the significance of the single-valued logarithm in complex analysis?

The single-valued logarithm is an important function in complex analysis as it allows for the manipulation and simplification of complex numbers in their polar form. It is also used in solving complex equations, finding complex roots, and evaluating complex integrals. Additionally, it plays a crucial role in understanding complex functions such as the complex exponential and trigonometric functions.

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