Is this manipulation "evil" or simply not intuitive due to unfamiliarity?

In summary, you should substitute the beta function when it has limits 0 and 1 and the question has limits 0 and infinity.
  • #1
JC2000
186
16
Homework Statement
Show that :
$$
\beta(p, q)=\int_{0}^{\infty} \frac{y^{q-1}}{(1+y)^{p+q}} d y
$$
Relevant Equations
The Beta Function :

$$
\beta(m, n)=\int_{0}^{1} x^{m-1}(1-x)^{n-1} d x
$$
The solution is as follows :
Screenshot 2020-06-09 at 9.11.35 PM.png

Screenshot 2020-06-09 at 9.12.16 PM.png
The substitution is what nags me , which is as follows :
Screenshot 2020-06-09 at 9.08.07 PM.png

This substitution "trick" to me seems impossibly difficult to arrive at "logically" without pretty much reverse engineering the problem.
So is this simply a lack of practice/ familiarisation showing ? I feel like I have always struggled with similar manipulations. In the past I think it was through repeatedly encountering a certain peculiar manipulation that it would become second nature for me. Is this how such problems are to be approached? Since it seems to almost be the same as memorising the solution!

Thank you for your insight!
 

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  • #2
Substitutions for indefinite integrals can be tricky and you pretty much have to learn them, however for definite integrals the question gives you a big clue - look at the limits.
 
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  • #3
pbuk said:
Substitutions for indefinite integrals can be tricky and you pretty much have to learn them, however for definite integrals the question gives you a big clue - look at the limits.

For indefinite integrals do you mean I should start listing the substitutions once they ambush me or is it fine to internalise them by practicing often?

So for this integral since the beta function has limits 0 and 1 while the question has limits 0 and infinity, I must search for a substitution whereby the desired transformation of limits is achieved (?)

Thanks for the input!
 
  • #4
JC2000 said:
For indefinite integrals do you mean I should start listing the substitutions once they ambush me or is it fine to internalise them by practicing often?
Practice, practice, practice :smile:
JC2000 said:
So for this integral since the beta function has limits 0 and 1 while the question has limits 0 and infinity, I must search for a substitution whereby the desired transformation of limits is achieved (?)
Yes - if you see ## \infty ## on one side of the transformation you know you are looking for a substitution where one limit goes to 1/0, if you see ## \pi ## on one side only you are looking for a trigonometric substitution etc. Don't forget to consider (as in this case) that the transformation might 'swap' the upper and lower limit.
 
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  • #5
This tip is on pg. 76 of Special Functions by Askey, Andrews, and Roy in the chapter on hypergeometric functions:

There are six linear fractional transformations that will map a set of three points to a permutation of the three points. For example, the set ##\left\{ 0,1,\infty\right\}## will be mapped to itself by the mappings

$$x\rightarrow \, x, \, 1-x, \, \tfrac{1}{x} , \, \tfrac{1}{1-x} , \, 1-\tfrac{1}{x} = \tfrac{x-1}{x} , \, \tfrac{1}{1-\tfrac{1}{x}} = \tfrac{x}{x-1}$$
 
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Related to Is this manipulation "evil" or simply not intuitive due to unfamiliarity?

1. What is the difference between manipulation and evil?

Manipulation refers to the act of influencing or controlling someone or something in a clever or dishonest way. Evil, on the other hand, is a moral concept that refers to actions or intentions that are considered morally wrong or harmful. While manipulation can be used for both good and bad purposes, evil is always seen as negative and harmful.

2. How can I determine if a manipulation is evil or simply not intuitive?

Determining whether a manipulation is evil or simply not intuitive can be difficult, as it often depends on the intentions and effects of the manipulation. If the manipulation is used to deceive or harm others, it can be considered evil. However, if the manipulation is used to achieve a positive outcome, such as persuading someone to make a healthier lifestyle choice, it may not be considered evil.

3. Is manipulation always intentional?

No, manipulation does not always have to be intentional. Sometimes, people may manipulate others without even realizing it. This can happen when someone uses subtle or indirect tactics to influence someone else's thoughts or behaviors, without being aware of their own actions.

4. Can manipulation be used for good purposes?

Yes, manipulation can be used for good purposes. For example, advertisers often use manipulation techniques to persuade people to buy a product or support a cause. In this case, the manipulation is not considered evil as it is used to achieve a positive outcome.

5. How can I protect myself from being manipulated?

To protect yourself from being manipulated, it is important to be aware of common manipulation tactics and to trust your instincts. Pay attention to the intentions and actions of others, and question anything that doesn't feel right. It is also helpful to educate yourself on the subject and to communicate openly and assertively with others.

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