Taylor Expansion of 1/(r-r'): Explained

In summary, the Taylor Expansion of 1/(r-r') is a representation of a complex function using a series of terms that approximate the function at a given point. It is useful for approximating the value of a function without having to evaluate it directly. The expansion is calculated using derivatives and the coefficients represent the rate of change of the function. However, there are limitations to using this method as the accuracy of the approximation depends on the number of terms used and in some cases, the series may not converge.
  • #1
captainjack2000
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Homework Statement


Could someone please explain how the taylor expansion of 1/(r-r') turns into
( 1/r+(r'.r)/r^3 + (3(r.r')^2-r^2r'^2)/2r^5 +...)


Homework Equations





The Attempt at a Solution

 
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  • #2
i think it should be [itex]\frac{1}{|\vec{r}-\vec{r'}|}[/itex] yes?

use the 3d taylor expansion formula

[itex]\phi(\vec{r}+\vec{a})=\sum_{n=0}^{\infty} \frac{1}{n!} (\vec{a} \cdot \nabla)^n \phi(\vec{r})[/itex] with [itex]\phi(\vec{r}+\vec{a})=\frac{1}{|\vec{r}-\vec{r'}|}[/itex]
 

1. What is the Taylor Expansion of 1/(r-r')?

The Taylor Expansion of 1/(r-r') is a mathematical representation of the function 1/(r-r') using a series of terms that approximate the function at a given point. It is a way to express a complex function in a simpler form that is easier to work with.

2. Why is the Taylor Expansion of 1/(r-r') useful?

The Taylor Expansion of 1/(r-r') is useful because it allows us to approximate the value of the function at a given point without having to evaluate the function directly. This can be helpful in situations where the function is difficult to evaluate or when we only need an estimate of the function's value.

3. How is the Taylor Expansion of 1/(r-r') calculated?

The Taylor Expansion of 1/(r-r') is calculated using a series of derivatives of the function at a given point. These derivatives are then multiplied by the appropriate coefficients and added together to form the series representation of the function.

4. What is the significance of the coefficients in the Taylor Expansion of 1/(r-r')?

The coefficients in the Taylor Expansion of 1/(r-r') represent the rate of change of the function at the given point. They determine the shape and behavior of the function and can be used to approximate the value of the function at nearby points.

5. Are there any limitations to using the Taylor Expansion of 1/(r-r')?

Yes, there are limitations to using the Taylor Expansion of 1/(r-r'). The accuracy of the approximation depends on the number of terms used in the series. As more terms are added, the approximation becomes more accurate. However, in some cases, the series may not converge or may require an infinite number of terms to accurately represent the function.

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