A integral about expotential function

In summary, the conversation was about calculating an integral using a trick involving squaring it and changing coordinates from cartesian to polar. The person asking the question was self-learning quantum mechanics and did not know what math skills were needed. The conversation ended with the suggestion to use standard integration for the final step.
  • #1
athrun200
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0

Homework Statement



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Homework Equations


Unknown

The Attempt at a Solution



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How to calculate this integral?
I have tried substitution and by parts. But they fail to get the answer.
 

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  • #2
There is a trick for this, square the integral:
[tex]
\left(\int_{-\infty}^{\infty}e^{-x^{2}}dx\right)^{2}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^{2}+y^{2})}dxdy
[/tex]
Now the idea is to make a change of co-ordinates from cartesian to polar...
 
  • #3
Are you taking quantum mechanics before knowing what a gaussian integral is ? Because you shouldn't...
 
  • #4
dextercioby said:
Are you taking quantum mechanics before knowing what a gaussian integral is ? Because you shouldn't...

Oh you are right.
In fact, I am self-learning quantum mechanics, and I don't know what maths skills I need to have.
 
  • #5
So move on from my explanation:
[tex]
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^{2}+y^{2})}dxdy=\int_{0}^{2\pi}\int_{0}^{\infty}re^{-r^{2}}drd\theta =2\pi\int_{0}^{\infty}re^{-r^{2}}dr
[/tex]
I think I will leave the last bit to you as it's standard integration.
 

1. What is an exponential function?

An exponential function is a mathematical function in the form of f(x) = ab^x, where a is a constant and b is the base. It is characterized by a constant rate of change, where the output value increases or decreases at a constant percentage over equal intervals of the input variable.

2. What is the purpose of taking the integral of an exponential function?

The integral of an exponential function allows us to find the area under the curve of the function. This is useful in many applications, such as calculating growth rates, compound interest, and decay rates.

3. How do you solve an integral of an exponential function?

To solve an integral of an exponential function, you can use the power rule, which states that the integral of x^n is equal to (x^(n+1))/(n+1) + C, where C is a constant. You can also use substitution or integration by parts depending on the complexity of the function.

4. Can an exponential function have a negative base?

Yes, an exponential function can have a negative base. However, when the base is negative, the function will oscillate between positive and negative values, making it more difficult to interpret and solve the integral.

5. What are some real-life applications of integrals involving exponential functions?

Integrals involving exponential functions have many real-world applications, such as in finance, physics, and biology. For example, in finance, integrals can be used to calculate compound interest and future value of investments. In physics, they are used to model radioactive decay and population growth. In biology, they can be used to describe the rate of enzyme reactions and the spread of diseases.

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