Logarithmic function, complex analysis, Laurent series expansion, isolated singularity are closely related to “log z Laurent series.” Log z Laurent series provides a representation of the logarithmic function in a complex domain around an isolated singularity. The Laurent series expansion allows for the expression of the function as an infinite sum of terms, each of which consists of a power of the independent variable z multiplied by a coefficient. This representation is particularly useful in complex analysis for studying the behavior of functions in the neighborhood of singularities.
Structure of Logarithmic Series Expansion (Log z Laurent Series)
The logarithmic series expansion, also known as the Log z Laurent series, provides a way to represent the natural logarithm function as a power series. This series is often used to approximate the natural logarithm function for complex values of z.
The general form of the Log z Laurent series is:
Log(z) = Σ (-1)^n-1 * z^n / n
where:
- z is a complex variable
- n is an integer
The series converges for all complex values of z except for z = 0.
The first few terms of the series are:
Log(z) = z - z^2 / 2 + z^3 / 3 - z^4 / 4 + ...
The convergence of the series is determined by the ratio test. The ratio test states that if the limit of the ratio of two consecutive terms is less than 1, then the series converges. For the Log z Laurent series, the ratio of two consecutive terms is:
lim (n -> ∞) |(-1)^n * z^n / n| / |(-1)^(n+1) * z^(n+1) / (n+1)| = lim (n -> ∞) |z| / (n+1) = 0
Since the limit of the ratio is 0, the series converges for all complex values of z except for z = 0.
The Log z Laurent series can be used to approximate the natural logarithm function for complex values of z. The accuracy of the approximation depends on the number of terms used in the series. The more terms used, the more accurate the approximation will be.
The following table shows the number of terms required to achieve a given accuracy for the Log z Laurent series approximation:
| Number of Terms | Accuracy |
|---|---|
| 10 | 10^-5 |
| 20 | 10^-10 |
| 30 | 10^-15 |
| 40 | 10^-20 |
Question 1: What is the Laurent series expansion of log z?
Answer: The Laurent series expansion of log z is the following:
log z = ∑_(n=1)^∞ [(-1)^(n+1)]/n * (z-1)^n / z
where z is a complex number.
Question 2: How do you determine the radius of convergence of the Laurent series expansion of log z?
Answer: The radius of convergence of the Laurent series expansion of log z is 1. This is because the series converges when |z-1| < 1 and diverges when |z-1| > 1.
Question 3: What are the applications of the Laurent series expansion of log z?
Answer: The Laurent series expansion of log z has many applications, including:
- Evaluating integrals
- Solving differential equations
- Studying the behavior of functions near singularities
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