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Suppose I have to explain the exponential function to someone. Then I would not say something like "Well, we want to extend the $a^n=a\cdot a\cdot a\cdot a.......$ definition to non-naturals. So we do this trick which preserves the properties and so on....".

Instead, what I would say is: 'We have many intuitive real-world problems where the rate of change of a quantity is proportional to its current value, say population." Using those real world problems, I would get to the exponential function as $\lim_{n\rightarrow \infty} (1+\frac{x}{n})^n$. Using this, I would further derive properties like $exp(x+y)=exp(x)exp(y)$. Then I would conclude that this otherwise very important function happens to be the same as repeated multiplication when the argument is a natural number.

I would just like to have a similar presentation of the gamma function. Is there any intuitive real world problem which can be used to motivate the gamme function?

Ryder Rude
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    My general advice to anybody who needs a special motivation to study something is: don't study it. Try gardening, nice flowers, tasty fruit or vegetables,... ;-) –  Sep 21 '20 at 12:44
  • The best motivation for the Gamma function is the functional equation for the Riemann zeta function, $$\zeta(s)=2^s\pi^{s-1}\sin(\pi s/2)\Gamma(1-s)\zeta(1-s)$$ https://en.wikipedia.org/wiki/Riemann_zeta_function#Riemann's_functional_equation – Gerry Myerson Sep 21 '20 at 13:12
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    @ProfessorVector My advice for you would be to stop giving advices as you are not good at it – Ryder Rude Sep 21 '20 at 13:41
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    @Gerry Myerson What are the odds the OP would consider the Riemann zeta function an "intuitive real world problem"? Whatever that means. –  Sep 21 '20 at 13:41
  • @GerryMyerson That's interesting but it's beyond my level. Can this function be motivated by some differential equation describing a physical phenomenon ? – Ryder Rude Sep 21 '20 at 13:42
  • It's a functional equation for the zeta function. The zeta function is tied to the deepest properties of the prime numbers. The prime numbers are as real world as anything. – Gerry Myerson Sep 22 '20 at 01:30
  • @RyderRude I do not want to start a debate whether there are physical applications. Only one advice : Do not trust blindly mathematical models for physics. Fascinating theories need not be real theories. Better focus on mathematical motivations. – Peter Sep 25 '20 at 16:03
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    Lots of interpretations here. – Simply Beautiful Art Sep 25 '20 at 17:06
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    @ProfessorVector you should delete your first comment. It is utter nonsense – mathworker21 Sep 25 '20 at 17:44
  • There is a nice motivation from Gauss sums. However, seeing how the OP responds to the zeta function comment, I hesitate to post an answer, as Gauss sums are probably not "real world" for the OP. – WhatsUp Sep 28 '20 at 09:44
  • @WhatsUp Your comment confuses me. The OP responded to the zeta function comment by saying "it's beyond my level"; it didn't seem he/she was saying it's not real world enough.... – mathworker21 Sep 30 '20 at 14:17
  • @mathworker21 I meant the response "motivated by some differential equation describing a physical phenomenon". The OP probably has background in physics/analysis etc., but not in number theory. And Gauss sums are certainly more on the number theory side. – WhatsUp Sep 30 '20 at 15:37
  • @WhatsUp thanks. my comment was stupid – mathworker21 Sep 30 '20 at 16:16

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Start with perhaps a discussion on mean time between failures and how it is used in engineering. Then talk about probability density function and how that describes rate of failures of components in a complex system and analysis of that requires a tool. That tool is the Gamma function, $\Gamma (x)$.

Problem: Suppose that a cell phone battery charge has the gamma distribution with shape parameter $k \in (0, \infty)$ and scale parameter $b \in (0, \infty)$. The probability distribution is given by

$$f(x) = {1 \over \Gamma (k) b^k}x^{k-1}e^{-x/b}, x \in (0, \infty)$$

Derive this formula and then calculate the probability that the battery charge will last more than 24 hours under what parameters $k$ and $b$. Ask them how this helps in the real world.

If you need more examples, you could talk about arrival times in Poisson processes, how Poisson processes have a gamma distribution. Most people should be able to relate to waiting in line in a queue or being put on hold by a call center. These are due to underlying gamma distributions. Modeling them requires using $\Gamma (x)$.

Problem: Suppose that customers arrive at a service station according to the Poisson model, at a rate of 10 per hour. Relative to a given starting time, find the probability that the second customer arrives sometime after 1 hour. Derive the formula and then link it back to gamma distributions.

Then you can take a short detour in explaining recursive processes and that is where the factorial comes in.

For integers $n! = n \times (n-1) \times \dots \times 2 \times 1$. The factorial function has a recursive property: $n! = n \times (n-1)!$.

What happens if $n$ is not integer or a real number or a complex number? It so happens that $\Gamma (x)$ is a function on complex numbers (which also covers reals) that behaves similar to how the factorial function behaves for integers.

$$\Gamma (x) = x \Gamma (x-1)$$

Notice how this is analagous to the factorial function.

But, what would be the definition of $\Gamma (x)$.

It so happens that $\Gamma (x)$ can be defined analytically as:

$$\Gamma (x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt $$

This function can be extended to negative real numbers and complex numbers as long as their real part $Re(x) \ge 1$.

You can then give a beautiful equation such as

$$\Gamma \left({1 \over 2}\right) = \sqrt \pi$$

and ask them to derive it.

If we just stop here, we can recap the concepts needed for understanding $\Gamma (x)$.

  1. Integers
  2. Factorial function
  3. Reals
  4. Complex numbers
  5. Integrals, areas under curves
  6. Analytical definition of $\Gamma (x)$
  7. Discussion on why $Re(x) \ge 1$
vvg
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