Estimate of a Principal Value Integral

In summary, a principal value integral is an integral that involves a singularity within the domain of integration and requires a special technique called the principal value method to be evaluated. This involves dividing the integral into two separate integrals and adding their values together. Principal value integrals have applications in physics, engineering, and mathematics, but can have limitations such as multiple singularities or lack of convergence in certain cases. Careful consideration of the function and singularity is important when using the principal value method.
  • #1
Euge
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For ##x\in \mathbb{R}##, let $$A(x) = \frac{1}{2\pi}\, P.V. \int_{-\infty}^\infty e^{i(xy + \frac{y^3}{3})}\, dy$$ Show that the integral defining ##A(x)## exists and ##|A(x)| \le M(1 + |x|)^{-1/4}## for some numerical constant ##M##.
 
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  • #2
I assume by [itex]A(x)[/itex] you mean [itex]\operatorname{Ai}(x)[/itex] or vice-versa.
 
  • #3
I meant ##A(x)##, sorry.
 

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