How to Fix Limits of Integration for Gamma Integral #6?

In summary, the conversation revolves around solving a gamma integral with unclear limits of integration. The speaker requests for help in fixing the limits, and the other person provides a solution using partial integration and the Gaussian error function. However, the speaker wants the solution to be expressed in terms of the gamma function, which the other person says can be done by using incomplete gamma functions. The conversation ends with the speaker expressing their difficulty with the topic and the other person acknowledging their level.
  • #1
AhmedHesham
96
11
Hi
I have a gamma integral in which it is not obvious how I can fix the limits of integration in order to match the standard form of gamma function.I just need someone to tell me how to fix them.
Screenshot_2021-05-02-20-33-03-14_e2d5b3f32b79de1d45acd1fad96fbb0f.jpg

I mean the integral number 6 in the picture.
You can see my attempt in the PDF .
 

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  • #2
[tex]I_6=\int_2^3 (3-y)^2 e^{-y^2}dy[/tex]

[tex]I_6=\frac{1}{2}\int_4^9 (z^{1/2}-6+9z^{-1/2}) e^{-z}dz[/tex]

By partial integration

[tex]\int_4^9 z^{1/2} e^{-z}dz=[-z^{1/2}e^{-z}]_4^9 + \frac{1}{2} \int_4^9 z^{-1/2}e^{-z}dz [/tex]

Gauss error function works to estimate the integral of RHS.
 
Last edited:
  • #3
anuttarasammyak said:
[tex]I_6=\int_2^3 (3-y)^2 e^{-y^2}dy[/tex]

[tex]I_6=\frac{1}{2}\int_4^9 (z^{1/2}-6+9z^{-1/2}) e^{-z}dz[/tex]

By partial integration

[tex]\int_4^9 z^{1/2} e^{-z}dz=[-z^{1/2}e^{-z}]_4^9 + \frac{1}{2} \int_4^9 z^{-1/2}e^{-z}dz [/tex]

Gauss error function works to estimate the integral of RHS.
Thanks for help
You didn't use the gamma function
The error function is not the subject of my course .
These exercises are on gamma function
Please make it in terms of gamma function.
 
  • #4
If you want to put the integrand in the form [itex]f(t)e^{-t}[/itex] then you are forced to take [itex]t = (x - 3)^2[/itex] and hence [tex]
\int_0^1 x^2 e^{-(x-3)^2}\,dx = \int_4^9 \frac{(3 - \sqrt {t})^2}{2 \sqrt{t}}e^{-t}\,dt.[/tex] That's not a sum of gamma functions, because the limits are 4 and 9 not 0 and [itex]\infty[/itex]. It can, however, be expresed as a sum of incomplete gamma functions.
 
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  • #5
Ok
I will ask my professor
Incomplete gamma function is beyond the scope of my course .
May be it was a mistake to give us it in the exercises on gamma function
 
  • #6
AhmedHesham said:
Please make it in terms of gamma function.
Incomplete gamma function and error function have the relation
[tex]\gamma(\frac{1}{2},x)=\sqrt{\pi}\ erf(\sqrt{x})[/tex]
 
  • #7
anuttarasammyak said:
Incomplete gamma function and error function have the relation
[tex]\gamma(\frac{1}{2},x)=\sqrt{\pi}\ erf(\sqrt{x})[/tex]
Ok
Thanks
You seem to be at a higher level.
 

1. What is a gamma integral?

A gamma integral is an integral that involves the gamma function, which is a special function that extends the factorial function to real and complex numbers. It is often used in mathematical analysis and statistics.

2. Why is help needed with Gamma Integral #6?

Gamma Integral #6 is a specific integral that may be difficult to solve or understand without prior knowledge or assistance. It is common for individuals to seek help with this integral in order to fully grasp its concept and solve it correctly.

3. How do you solve Gamma Integral #6?

The solution to Gamma Integral #6 involves using integration by parts, substitution, or other techniques depending on the specific integral. It is important to have a strong understanding of the properties of the gamma function and integration methods in order to solve this integral accurately.

4. What are the applications of Gamma Integral #6?

Gamma Integral #6 is commonly used in various fields such as physics, engineering, and economics to solve problems related to probability, statistics, and differential equations. It can also be used to evaluate certain series and products.

5. Are there any online resources for help with Gamma Integral #6?

Yes, there are various online resources such as math forums, tutorial websites, and online calculators that can provide assistance with solving Gamma Integral #6. It is important to use reliable sources and to understand the steps and methods used in the solution.

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