Integration of exponential function

In summary, the integral from negative infinity to positive infinity of the exponential function with a cubic polynomial in the exponent can be rewritten in terms of the Airy function, as shown in O.Vallee's book "Airy Functions and Applications to Physics." However, the formula is not proven in the book and the individual asking for hints on how to prove it has not been successful.
  • #1
jshw
4
0

Homework Statement


Int(-infinity to +infinity) exp[i(t^3/3 + at^2 + bt)]dt = 2pi*exp[ia(2a^2/3 - b)]*Ai(b-a^2)
O.Vallee gives this formula in his book, "Airy Functions and Applications to Physics"
but there are no proof of this formula. I tried to prove this formula, but I failed.
Would you give some hints?


Homework Equations


Int(-infinity to +infinity) exp[i(t^3/3 + at^2 + bt)]dt = 2pi*exp[ia(2a^2/3 - b)]*Ai(b-a^2)


The Attempt at a Solution

 
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  • #2
First I have to rewrite it in nicer notation:
[tex]\int_{-\infty}^{+\infty} \exp\left[i\left(\frac{t^3}{3} + at^2 + bt\right)\right]\mathrm{d}t = 2\pi \exp\left[ia\left(\frac{2a^2}{3} - b\right)\right]\mathrm{Ai}(b-a^2)[/tex]
Now that that's taken care of... surely you must have a definition of the Airy function to work with? What is it? (That's what you should have put in the "relevant equations" section)
 

1. What is the integration of an exponential function?

The integration of an exponential function involves finding the antiderivative of the function, which is the original function before it was differentiated.

2. How do you integrate an exponential function?

To integrate an exponential function, you can use the power rule, which states that the integral of x^n is (x^(n+1))/(n+1). For an exponential function, the exponent will become the power of the base and the coefficient will become the constant of integration.

3. What is the general formula for integrating exponential functions?

The general formula for integrating exponential functions is ∫a^x dx = (a^x)/ln(a) + C, where a is the base of the exponential function and C is the constant of integration.

4. Can you give an example of integrating an exponential function?

Yes, for example, to integrate the function f(x) = 2^x, we can use the formula above to get ∫2^x dx = (2^x)/ln(2) + C.

5. Why is it important to understand integration of exponential functions?

Understanding integration of exponential functions is important in many areas of science, particularly in calculus and physics. It allows us to find the area under the curve and to solve problems involving exponential growth or decay.

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