How to handle the large $r$ limit of this integral?

In summary, The person is stuck with an integral of the form lim_{r \to \infty} \int_{-1}^1 dt f(t) e^{i r (t-1)} for some unknown function f(t). They are wondering if there is a way to solve it using the saddle-point method, but it is not applicable in this case since the argument of the exponential does not have a minimum. They ask for advice on how to approach the integral and suggest that it may be dominated by a certain value of t, but the other person mentions that the "method of stationary phase" may not work in this situation. They suggest checking out Wikipedia for more information.
  • #1
praharmitra
311
1
I am doing some calculation and am now stuck with an integral of the form

[tex] \lim_{r \to \infty} \int_{-1}^1 dt f(t) e^{i r (t-1)} [/tex]

for some function [itex]f(t)[/itex]. I don't know what the exact form of [itex]f(t)[/itex] is.

Is there any way to address this integral? Similar to the saddle-point method perhaps? The saddle-point method does not work here right? since the argument of the exponential does not have a minima.

How should I go about this?

Can we say that this integral is dominated by a certain value of [itex]t[/itex], say at [itex]t=1[/itex]? Why or why not?
 
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  • #2
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FAQ: How to handle the large $r$ limit of this integral?

What is the large $r$ limit of an integral and why is it important in science?

The large $r$ limit of an integral refers to the behavior of the integral as the upper limit of integration ($r$) approaches infinity. This limit is important in science because it allows us to understand the long-term behavior of physical systems and make predictions about their behavior at very large scales.

How is the large $r$ limit of an integral calculated?

The large $r$ limit of an integral can be calculated using various techniques such as substitution, integration by parts, or using properties of special functions. The specific method used will depend on the form of the integral and the problem being solved.

What is the significance of the large $r$ limit in quantum mechanics?

In quantum mechanics, the large $r$ limit of an integral is used to describe the long-range behavior of interactions between particles. This is important in understanding the behavior of atoms and molecules, as well as in describing the properties of materials and chemical reactions.

Can the large $r$ limit of an integral be used to solve real-world problems?

Yes, the large $r$ limit of an integral has many practical applications in various fields of science and engineering. For example, it can be used to calculate the electric potential at large distances from a point charge, or to describe the behavior of electromagnetic waves in free space.

Are there any limitations to using the large $r$ limit of an integral?

While the large $r$ limit of an integral is a useful tool, it may not always accurately describe the behavior of a physical system at very large scales. In some cases, the behavior of the system may be better understood using other mathematical techniques such as asymptotic analysis or numerical methods.

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