Orthogonal Functions | Homework Statement

In summary, the conversation discusses finding an antiderivative for a given function and using this to evaluate an integral. The extra condition is that the function must be continuous, and there is a discussion about the properties of the resulting antiderivative function.
  • #1
dirk_mec1
761
13

Homework Statement




http://img55.imageshack.us/img55/8494/67023925dy7.png [Broken]



The Attempt at a Solution



All functions orthogonal to 1 result in the fact that: [tex] \int_a^b f(t)\ \mbox{d}t =0 [/tex]

Now the extra condition is that f must be continous. (because of the intersection).

But where does the fact that f(a)=f(b)=0 comes from? And why look at the deratives?
 
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  • #2
Remember way back in first year calc, when you learned that to do that integral you find an antiderivative F(x) and evaluate F(b)-F(a). This is that same problem in disguise.
 
  • #3
Well I thought of this: [tex] \int_a^b \int_a^t f(s)\ \mbox{d}s \mbox{d}t =0[/tex]
 
  • #4
dirk_mec1 said:
Well I thought of this: [tex] \int_a^b \int_a^t f(s)\ \mbox{d}s \mbox{d}t =0[/tex]

Fine. What are you going to do with it? Why don't you just define [tex]
F(x)=\int_a^x f(s)\ \mbox{d}s
[/tex]
What are some of the properties of F(x)?
 

1. What are orthogonal functions?

Orthogonal functions are a set of mathematical functions that are perpendicular to each other when plotted on a graph. This means that the integral of their product over a certain interval is equal to zero, making them independent of each other.

2. How are orthogonal functions used in science?

Orthogonal functions are used in many areas of science, such as physics, engineering, and statistics. They are commonly used in Fourier series and Fourier transforms to represent complex functions and signals. They also have applications in solving differential equations and in data analysis.

3. What is the significance of orthogonality in orthogonal functions?

The orthogonality of these functions allows for simplification and easier manipulation in mathematical calculations. It also allows for the representation of complex functions as a sum of simpler components, making it easier to analyze and understand them.

4. What is the difference between orthogonal and orthonormal functions?

Orthogonal functions are perpendicular to each other, while orthonormal functions not only have this property but also have a magnitude of 1. This means that the integral of the product of orthonormal functions over an interval is equal to 1, making them a more specialized subset of orthogonal functions.

5. Can you give an example of orthogonal functions?

One example of orthogonal functions is the sine and cosine functions. When plotted on a graph, these functions are perpendicular to each other and their product has an integral of 0 over a certain interval. Other examples include Legendre polynomials, Bessel functions, and Chebyshev polynomials.

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