Math Problem Statement

Series ln((L/2+sqrt(x^2+L^2/4))/(-L/2+sqrt(x^2+L^2/4)))/(Lx) for x=0

Solution

To find the series expansion of the given expression

ln(L2+x2+L24L2+x2+L24)Lx\frac{\ln\left(\frac{\frac{L}{2} + \sqrt{x^2 + \frac{L^2}{4}}}{-\frac{L}{2} + \sqrt{x^2 + \frac{L^2}{4}}}\right)}{Lx}

around x=0x = 0, we need to carefully analyze the behavior of the function as xx approaches 0. Here are the steps:

1. Simplifying the Expression:

We begin by simplifying the expression inside the logarithm. The form is quite complicated, but it can be tackled step by step.

L2+x2+L24andL2+x2+L24.\frac{L}{2} + \sqrt{x^2 + \frac{L^2}{4}} \quad \text{and} \quad -\frac{L}{2} + \sqrt{x^2 + \frac{L^2}{4}}.

For small xx, we will expand the square root expression using a Taylor series.

2. Approximate the Square Root:

For small xx, we approximate x2+L24\sqrt{x^2 + \frac{L^2}{4}}. Using the binomial expansion for small xx, we get:

x2+L24L2(1+2x2L2)1/2L2(1+x2L2).\sqrt{x^2 + \frac{L^2}{4}} \approx \frac{L}{2} \left(1 + \frac{2x^2}{L^2}\right)^{1/2} \approx \frac{L}{2} \left(1 + \frac{x^2}{L^2}\right).

3. Substitute this into the expression:

Now substitute this approximation into the original formula for the numerator:

L2+x2+L24L2+x2+L24.\frac{\frac{L}{2} + \sqrt{x^2 + \frac{L^2}{4}}}{-\frac{L}{2} + \sqrt{x^2 + \frac{L^2}{4}}}.

This will give a series in terms of xx, which can then be used to find the logarithm.

4. Logarithmic Expansion:

The logarithmic function can be expanded as a series for small values of xx, which is often done using:

ln(1+z)zfor smallz.\ln(1 + z) \approx z \quad \text{for small} \quad z.

After substituting the approximation of the square root, simplify the entire expression inside the logarithm. This will give a leading term plus higher-order terms in xx.

5. Divide by LxLx:

Finally, divide the resulting series by LxLx to find the series expansion for the whole expression.


Final Result (Leading Term):

The leading term of the expansion for the expression is expected to be of order 1x\frac{1}{x}, with additional terms involving higher powers of xx.


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Math Problem Analysis

Mathematical Concepts

Series Expansion
Logarithmic Functions
Square Root Approximation
Binomial Expansion

Formulas

ln(1 + z) ≈ z for small z
Binomial Expansion: (1 + x)^n ≈ 1 + nx for small x

Theorems

Taylor Series Expansion
Logarithmic Approximation for Small Values

Suitable Grade Level

College/Undergraduate