# Product of complex conjugate functions with infinite sums

• I
In summary, the conversation discusses a complex function involving a sum and a Legendre polynomial. The author provides a second equation that is derived from the first, using two different indices to include cross-terms. The conversation ends with a thank you to the expert summarizer.
Hello there. I'm here to request help with mathematics in respect to a problem of quantum physics. Consider the following function $$f(\theta) = \sum_{l=0}^{\infty}(2l+1)a_l P_l(cos\theta) ,$$ where ##f(\theta)## is a complex function ##P_l(cos\theta)## is the l-th Legendre polynomial and ##a_l## is a the complex term (known as partial wave amplitude). Then, the author calculates the product of ##f(\theta)## with its complex conjugate, and shows the result: $$f(\theta)f^{*}(\theta) = |f(\theta)|^2 = \sum_{l}\sum_{l'}(2l+1)(2l'+1)(a_l)^{*}(a_{l'})P_l(cos\theta)P_{l'}(cos\theta).$$ My problem here is to understand how he obtained the second equation. I'm not familiar with operations with sums and real analysis, and I'm stuck with it. Any help will be very appreciated.

Which part is unclear? The second equation directly follows from the first. The sums work like with real numbers. The complex conjugation is only relevant for the complex amplitude, where you also find it in the second equation.

mfb said:
Which part is unclear? The second equation directly follows from the first. The sums work like with real numbers. The complex conjugation is only relevant for the complex amplitude, where you also find it in the second equation.
Good pointed; more precisely, I do not understand why he chooses a different index for each sum.

Let's make an example, with a sum over just three terms: ##(x_1 + x_2 + x_3) \cdot (y_1 + y_2 + y_3)##. This will lead to 9 terms: ##x_i y_j## for i=1,2,3, and j=1,2,3 separately. As formula, ##(\sum_{i=1}^{3}x_i) (\sum_{i=1}^{3} y_i) = (x_1 + x_2 + x_3) \cdot (y_1 + y_2 + y_3) = (x_1 y_1 + x_1 y_2 + x_1 y_3 + x_2 y_1 + x_2 y_2 + x_2 y_3 + x_3 y_1 + x_3 y_2 + x_3 y_3) = \sum_{i=1}^{3} \sum_{j=1}^{3} x_i y_j##. You need two different indices to have to cross-terms (like ##x_1 y_3##) included.

mfb said:
Let's make an example, with a sum over just three terms: ##(x_1 + x_2 + x_3) \cdot (y_1 + y_2 + y_3)##. This will lead to 9 terms: ##x_i y_j## for i=1,2,3, and j=1,2,3 separately. As formula, ##(\sum_{i=1}^{3}x_i) (\sum_{i=1}^{3} y_i) = (x_1 + x_2 + x_3) \cdot (y_1 + y_2 + y_3) = (x_1 y_1 + x_1 y_2 + x_1 y_3 + x_2 y_1 + x_2 y_2 + x_2 y_3 + x_3 y_1 + x_3 y_2 + x_3 y_3) = \sum_{i=1}^{3} \sum_{j=1}^{3} x_i y_j##. You need two different indices to have to cross-terms (like ##x_1 y_3##) included.
Oh, it's very clear now! Thank you so much :)

mfb

## 1. What is the definition of a product of complex conjugate functions with infinite sums?

The product of complex conjugate functions with infinite sums is a mathematical operation that involves multiplying two functions that are complex conjugates of each other, while also taking the limit of the sum of these functions as it approaches infinity.

## 2. How do you solve a product of complex conjugate functions with infinite sums?

To solve a product of complex conjugate functions with infinite sums, you first need to simplify the product of the two functions. Then, you can use the properties of infinite sums and limits to evaluate the limit of the sum as it approaches infinity.

## 3. What are the applications of products of complex conjugate functions with infinite sums?

Products of complex conjugate functions with infinite sums are commonly used in physics and engineering to model systems with oscillating behavior, such as electrical circuits and mechanical vibrations. They are also used in signal processing and wave analysis.

## 4. Can products of complex conjugate functions with infinite sums have imaginary components?

Yes, products of complex conjugate functions with infinite sums can have imaginary components. In fact, the imaginary components are often necessary to accurately represent the behavior of oscillating systems.

## 5. How does changing the functions in the product affect the result of a product of complex conjugate functions with infinite sums?

Changing the functions in the product can greatly affect the result of a product of complex conjugate functions with infinite sums. It can change the amplitude, frequency, and phase of the oscillations, which can have significant implications in the applications of this mathematical operation.

• General Math
Replies
15
Views
2K
Replies
16
Views
1K
• Electromagnetism
Replies
2
Views
908
• General Math
Replies
1
Views
5K
• Calculus
Replies
2
Views
1K
• Quantum Physics
Replies
0
Views
531
• General Math
Replies
4
Views
2K
• Quantum Physics
Replies
14
Views
3K
Replies
10
Views
2K
• Precalculus Mathematics Homework Help
Replies
10
Views
2K