Infinite Product: Showing & Evaluating

In summary: Use the Weierstrass factorization theorem.In summary, we discussed the infinite product formula $\displaystyle \prod_{k=1}^{\infty} \left( 1- \frac{z^{n}}{k^{n}} \right) = \prod_{k=0}^{n-1} \frac{1}{\Gamma\left[ 1-\exp (2 \pi i k/n) z\right]}$ and how it can be used to show the relationship between the infinite product $\displaystyle \prod_{k=1}^{\infty} \left(1- \frac{z^{2}}{k^{2}} \right)$ and the sine function. We
  • #1
polygamma
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1) Show that for $n >1$, $\displaystyle \prod_{k=1}^{\infty} \left( 1- \frac{z^{n}}{k^{n}} \right) = \prod_{k=0}^{n-1} \frac{1}{\Gamma\left[ 1-\exp (2 \pi i k/n) z\right]}$.2) Use the above formula to show that $ \displaystyle \prod_{k=1}^{\infty} \left(1- \frac{z^{2}}{k^{2}} \right) = \frac{\sin \pi z}{\pi z}$.

3) Evaluate $ \displaystyle \prod_{k=2}^{\infty} \left(1- \frac{1}{k^{3}} \right)$.
 
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  • #2
Re: infinite product

Random Variable said:
1) Show that for $n >1$, $\displaystyle \prod_{k=1}^{\infty} \left( 1- \frac{z^{n}}{k^{n}} \right) = \prod_{k=0}^{n-1} \frac{1}{\Gamma\left[ 1-\exp (2 \pi i k/n) z\right]}$.2) Use the above formula to show that $ \displaystyle \prod_{k=1}^{\infty} \left(1- \frac{x^{2}}{k^{2}} \right) = \frac{\sin \pi z}{\pi z}$.

3) Evaluate $ \displaystyle \prod_{k=2}^{\infty} \left(1- \frac{1}{k^{3}} \right)$.

Are you reading Serge Lang's book?
 
  • #3
Re: infinite product

@dwmsmith

I don't know who that is.
 
  • #5
Re: infinite product

I thought I had put together three interesting questions since I've never seen $\displaystyle \prod_{k=1}^{\infty} \left(1- \frac{z^{2}}{k^{2}} \right) = \frac{\sin \pi z}{\pi z}$ derived from that formula.And to evaluate the second product using that formula requires a bit of a trick.(Sadface)
 
  • #6
Re: infinite product

Random Variable said:
I thought I had put together three interesting questions since I've never seen $\displaystyle \prod_{k=1}^{\infty} \left(1- \frac{z^{2}}{k^{2}} \right) = \frac{\sin \pi z}{\pi z}$ derived from that formula.And to evaluate the second product using that formula requires a bit of a trick.(Sadface)

It happened a lot for me last days . Nevertheless , I had interesting time deriving ''what I thought" new formulas .
 
  • #7
Re: infinite product

Random Variable said:
I thought I had put together three interesting questions since I've never seen $\displaystyle \prod_{k=1}^{\infty} \left(1- \frac{z^{2}}{k^{2}} \right) = \frac{\sin \pi z}{\pi z}$ derived from that formula.And to evaluate the second product using that formula requires a bit of a trick.(Sadface)

Part 3 isn't there or I can't find it.
 
  • #8
Re: infinite product

ZaidAlyafey said:
It happened a lot for me last days . Nevertheless , I had interesting time deriving ''what I thought" new formulas .

I knew it wasn't a new formula. It's listed on Wolfram MathWorld with a reference to a book from 1986. And I'm sure it was known well before then. I had just never seen it used anywhere to evaluate the infinite product for $\sin$ or anything else.
 
  • #9
Re: infinite product

dwsmith didn't post the solution he said he had for the first problem, so I'm going to at least post that.Euler's limit defintion of the gamma function is $ \displaystyle \Gamma(z) = \lim_{m \to \infty} \frac{m! \ m^{z}}{z(z+1) \cdots (z+m)} $.

So $\displaystyle \Gamma (1+z) = z \Gamma (z) = \lim_{m \to \infty} \frac{m! \ m^{z}}{(z+1) \cdots (z+m)} = \lim_{m \to \infty} m^{z} \prod_{k=1}^{m} \Big(1+ \frac{z}{k} \Big)^{-1}$$\displaystyle \Gamma \big[ 1-\exp(2 \pi i l /n)z \big] = \lim_{m \to \infty} m^{-\exp(2 \pi i l /n)z} \prod_{k=1}^{m} \Big(1- \frac{\exp(2 \pi i l/n)z}{k} \Big)^{-1}$

$ \displaystyle \prod_{l=0}^{n-1} \Gamma \big[1-\exp(2 \pi i l/n)z \big] = \lim_{m \to \infty} m^{- \sum_{l=0}^{n-1} \exp(2 \pi i l/n)z} \prod_{l=0}^{n-1} \prod_{k=1}^{m} \Big(1- \frac{\exp(2 \pi i l/n)z}{k} \Big)^{-1}$

$ \displaystyle = \lim_{m \to \infty} \prod_{k=1}^{m} \prod_{l=0}^{n-1} \Big(1- \frac{\exp(2 \pi i l/n)z}{k} \Big)^{-1}$And since $\displaystyle x^n-z = \prod_{k=0}^{n-1} \Big( x- z^{\frac{1}{n}} \exp(2 \pi i k /n) \Big) \implies 1- z^{n} = \prod_{k=0}^{n-1} \Big( 1- \exp(2 \pi i k /n)z\Big)$,

$ \displaystyle \lim_{m \to \infty} \prod_{k=1}^{m} \prod_{l=0}^{n-1} \Big(1- \frac{\exp(2 \pi i l/n)z}{k} \Big)^{-1} = \lim_{m \to \infty} \prod_{k=1}^{m} \Big( 1- \frac{z^{n}}{k^{n}} \Big) ^{-1} = \prod_{k=1}^{\infty} \left(1 - \frac{z^{n}}{k^{n}} \right)^{-1}$
Then for the second question, $\displaystyle \prod_{k=1}^{\infty} \left( 1- \frac{z^{2}}{k^{2}} \right) = \frac{1}{\Gamma(1-z) \Gamma(1+z)} = \frac{1}{z \Gamma(1-z) \Gamma(z)} = \frac{\sin \pi z}{\pi z}$Hint for the third question:

$ \displaystyle \prod_{k=2}^{\infty} \left( 1- \frac{1}{k^{3}} \right) = \lim_{z \to 1} \frac{1}{1-z^{3}} \prod_{k=1}^{\infty} \left(1 - \frac{z^{3}}{k^{3}} \right) $
 

FAQ: Infinite Product: Showing & Evaluating

1. What is an infinite product?

An infinite product is an expression that involves multiplying an infinite number of terms together. It is the counterpart to an infinite sum, which involves adding an infinite number of terms together.

2. How do you show an infinite product?

To show an infinite product, you must write out the terms of the product and then take the limit as the number of terms approaches infinity. This means that you must evaluate the product at various values and observe any patterns that emerge.

3. What is the difference between showing and evaluating an infinite product?

Showing an infinite product involves writing out the terms and taking the limit to demonstrate that the product converges or diverges. Evaluating an infinite product involves finding the actual numerical value of the product, either through a formula or by using a computer or calculator.

4. How do you determine if an infinite product converges or diverges?

To determine if an infinite product converges or diverges, you can use various tests such as the ratio test, the root test, or the integral test. These tests compare the infinite product to known convergent or divergent series and can help determine the behavior of the product.

5. What are some real-world applications of infinite products?

Infinite products are used in various fields of science, such as physics, chemistry, and biology. They are used to model exponential growth, calculate probabilities, and approximate solutions to differential equations. They are also used in financial mathematics to model compound interest and in engineering to design efficient systems.

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