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liyz06
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Homework Statement
I'm reading Hinch's perturbation theory book, and there's a statement in the derivation:
...[itex]\int_z^{\infty}\dfrac{d e^{-t^2}}{t^9}<\dfrac{1}{z^9}\int_z^{\infty}d e^{-t^2}[/itex]...
Why is that true?
liyz06 said:Homework Statement
I'm reading Hinch's perturbation theory book, and there's a statement in the derivation:
...[itex]\int_z^{\infty}\dfrac{d e^{-t^2}}{t^9}<\dfrac{1}{z^9}\int_z^{\infty}d e^{-t^2}[/itex]...
Why is that true?
Homework Equations
The Attempt at a Solution
Dick said:Because 1/t^9 for t in (z,infinity) is less than 1/z^9. Draw a graph.
A Gaussian integral is a mathematical integral of the form ∫e-x2dx, where e is the base of the natural logarithm and x is the variable of integration. It is also known as the error function and is used to calculate the area under a bell-shaped curve.
Gaussian integrals have many applications in statistics, physics, and engineering. They are commonly used to solve problems involving probability distributions, heat transfer, and quantum mechanics. They also have practical applications in signal processing and image processing.
To solve a Gaussian integral, you can use integration by parts or the substitution method. It is also possible to use special tables or software programs to calculate the value of the integral. Additionally, there are various approximation methods that can be used for more complex integrals.
The Gaussian integral is closely related to the normal distribution, also known as the Gaussian distribution. The integral is used to calculate the probability of a given event occurring within a certain range of values in a normal distribution. In fact, the normal distribution is often referred to as the Gaussian curve due to its close connection to the Gaussian integral.
Yes, there are many real-world applications of Gaussian integrals. Some examples include calculating the probability of extreme weather events, analyzing financial data to predict stock market trends, and developing computer algorithms for data compression and image recognition. They are also used in various fields of engineering, such as designing efficient heating and cooling systems.