Convergence of Saddle-Point Approximation for Large M in Integrals

  • Thread starter Thread starter jfitz
  • Start date Start date
  • Tags Tags
    Approximation
AI Thread Summary
The discussion focuses on the applicability of the saddle-point approximation method for integrals of the form ∫ exp[M f(x) + g(x)]dx as M approaches infinity. The original query highlights a concern about the method's effectiveness when only part of the function becomes large, rather than the entire function being multiplied by a large number. The user ultimately resolves their question independently, indicating that the method can indeed be applied in this scenario. This suggests that the saddle-point approximation is versatile enough to handle integrals with mixed growth behaviors in their components. The conclusion reinforces the method's relevance in advanced integral analysis.
jfitz
Messages
11
Reaction score
0
Can the method of steepest descent (saddle point method) be used if an integral has the following form:

\int exp\left[M f(x) + g(x)\right]dx

where M goes to infinity?

I ask because all the examples I've seen of this method involve a function which is multiplied by a very large number, but never with only part of the function getting big.
 
Mathematics news on Phys.org
Nevermind, I figured it out.
 

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »
Thread 'Imaginary Pythagorus'

Thread 'Imaginary Pythagorus'

I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
View full post »

Similar threads

A Saddle point Integrals with logarithm and cosine
Replies
1
Views
2K
I Why does the integral of sine of x^2 from - infinity to + infinity diverge?
Replies
4
Views
2K
A How do I apply the method of steepest descents to this integral?
Replies
1
Views
2K
MHB Understanding Gauss Quadrature for Approximating Integrals
Replies
2
Views
2K
Steepest descent vs. stationary phase method
Replies
4
Views
4K
I Understanding different Monte Carlo approximation notations
Replies
7
Views
2K
MHB Fixed point iteration: g is a contraction mapping
Replies
17
Views
3K
MHB Fixed point,, Jacobi- & Newton Method, Linear Systems
Replies
7
Views
2K
I Optimization problem with multiple outputs: impossible?
Replies
6
Views
2K
I Landscape Fabric Roll
Replies
8
Views
3K

Hot Threads

Recent Insights

Back
Top