Evaluating scalar products of two functions

In summary, the task is to find the scalar product of the basis function ##\phi_i(x)## and the functions ##f(x)## and ##g(x)##, which can be done simply by multiplying the functions and evaluating the integral.
  • #1
guyvsdcsniper
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Homework Statement
Find ##\braket{f|f} , \braket{f|g} , \braket{g|g}##.
Relevant Equations
integral, scalar product
I am to consider a basis function ##\phi_i(x)##, where ##\phi_0 = 1 ,\phi_1 = cosx , \phi_2 = cos(2x) ## and where the scalar product in this vector space is defined by ##\braket{f|g} = \int_{0}^{\pi}f^*(x)g(x)dx##

The functions are defined by ##f(x) = sin^2(x)+cos(x)+1## and ##g(x) = cos^2(x)-cos(x)##
My task is to find ##\braket{f|f} , \braket{f|g} , \braket{g|g}##.

Given that f(x) and g(x) have no imaginary parts, is this problem really as simple as multiplying these two functions and evaluating the integral? If so I have no trouble doing that, I guess I just want to make sure I understand correctly what the question is asking. I am pretty sure my thought process is correct but I am just a bit unsure because of how simple the approach is.
ψ
 
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  • #2
guyvsdcsniper said:
Homework Statement:: Find ##\braket{f|f} , \braket{f|g} , \braket{g|g}##.
Relevant Equations:: integral, scalar product

I am to consider a basis function ##\phi_i(x)##, where ##\phi_0 = 1 ,\phi_1 = cosx , \phi_2 = cos(2x) ## and where the scalar product in this vector space is defined by ##\braket{f|g} = \int_{0}^{\pi}f^*(x)g(x)dx##

The functions are defined by ##f(x) = sin^2(x)+cos(x)+1## and ##g(x) = cos^2(x)-cos(x)##
My task is to find ##\braket{f|f} , \braket{f|g} , \braket{g|g}##.

Given that f(x) and g(x) have no imaginary parts, is this problem really as simple as multiplying these two functions and evaluating the integral? If so I have no trouble doing that, I guess I just want to make sure I understand correctly what the question is asking. I am pretty sure my thought process is correct but I am just a bit unsure because of how simple the approach is.
ψ
In four words, you have it right! :)

-Dan
 
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Related to Evaluating scalar products of two functions

1. What is a scalar product?

A scalar product, also known as a dot product, is a mathematical operation that takes two vectors and produces a single scalar value. It is commonly used in physics and engineering to calculate the magnitude of one vector in the direction of another vector.

2. How do you evaluate a scalar product of two functions?

To evaluate a scalar product of two functions, you first need to find the dot product of the two vectors that represent the functions. This can be done by multiplying the corresponding components of each vector and then summing the results. The resulting scalar value is the evaluated scalar product.

3. What is the significance of evaluating scalar products of two functions?

Evaluating scalar products of two functions is important in many fields of science and engineering. It can be used to calculate work done by a force, determine the angle between two vectors, and find the projection of one vector onto another. It also has applications in signal processing and statistics.

4. Can scalar products of two functions be negative?

Yes, scalar products of two functions can be negative. This occurs when the angle between the two vectors is greater than 90 degrees. In this case, the dot product will be negative, indicating that the two vectors are pointing in opposite directions.

5. Are there any properties or rules for evaluating scalar products of two functions?

Yes, there are several properties and rules that can be used when evaluating scalar products of two functions. These include the commutative property, distributive property, and the fact that the scalar product of a vector with itself is equal to the square of its magnitude. These properties can make the evaluation process more efficient and can also help with solving complex problems involving scalar products.

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