Proving Integral Equation: f(x)= O(x)

In summary, the conversation discusses the functions f(x) and g(x) defined by an integral equation and the desire to prove that f(x) is of order O(x). A change of variable yx=t is used to simplify the equation and it is shown that if the last integral exists, then there is a constant A such that x^{2}g(x) \le A for K(t) always being positive on the interval [0,oo). The order of g or K is necessary to make a definitive conclusion.
  • #1
zetafunction
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0
let be the functions f(x) and g(x) defined by an integral equation

[tex] g(x)= \int_{0}^{\infty}dy K(yx)f(y)dy [/tex]

then i want to prove that for example [tex] f(x)= O(x) [/tex]

then using a change of varialbe yx=t i manage to put

[tex] g(x) \le \frac{C}{x^{2}} \int_{0}^{\infty}dtK(t)t [/tex]

if the last integral exists , then a simple condition is that there will be a constant so [tex]x^{2} g(x) \le A [/tex] i assume K(t) is ALWAYS positive on the interval [0,oo)
 
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  • #2
It's difficult to decide unless one knows the order of g or K.
 

1. How do you prove that an integral equation is equal to O(x)?

To prove that an integral equation is equal to O(x), you need to show that the function f(x) is bounded by a multiple of x for large values of x. This can be done using the limit definition of big O notation and by finding a constant C such that |f(x)| ≤ Cx for all x ≥ x0, where x0 is some positive number.

2. What is the significance of proving an integral equation is equal to O(x)?

Proving that an integral equation is equal to O(x) shows that the function f(x) has a finite upper bound when x becomes large. This means that the function does not grow too quickly and can be approximated by a linear function for large values of x. This is useful in analyzing the behavior of functions and understanding their growth rates.

3. Can an integral equation be equal to both O(x) and o(x)?

Yes, an integral equation can be equal to both O(x) and o(x). This means that the function f(x) has a finite upper bound and also approaches zero as x becomes large. In other words, the function grows at a slower rate than a linear function and eventually becomes negligible compared to x.

4. What are some common techniques for proving an integral equation is equal to O(x)?

Some common techniques for proving that an integral equation is equal to O(x) include using the limit definition of big O notation, using the properties of integrals, and using algebraic manipulations to simplify the equation. Other techniques may also involve using mathematical induction, substitution, or comparison to other known functions.

5. How does proving an integral equation is equal to O(x) relate to the concept of convergence?

Proving that an integral equation is equal to O(x) is related to the concept of convergence because it shows that the function f(x) converges to a finite value as x becomes large. In other words, as x approaches infinity, the function does not continue to grow without bound but instead approaches a finite limit. This is important in analyzing the behavior of functions and determining their convergence or divergence.

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