Question

How to verify this equation $\prod_{j=1}^\infty{1+\exp(i2^{-j}\omega)\over2}={1-\exp(i\omega)\over i\omega}$?

score 121 · Answer 1 · 2013-04-08 20:01:12

121

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How to verify this equation $\prod_{j=1}^\infty{1+\exp(i2^{-j}\omega)\over2}={1-\exp(i\omega)\over i\omega}$?

Published on 08 Apr 2013 - 20:01

score 1.3k · Answer 2 · 2013-04-12 17:13:31

1.3k

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Infinite Product $\prod_{n=1}^\infty\left(1+\frac1{\pi^2n^2}\right)$

Published on 12 Apr 2013 - 17:13

score 3.1k · Answer 3 · 2013-04-15 23:48:31

3.1k

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Convergence of $\prod_{n=1}^\infty(1+a_n)$

Published on 15 Apr 2013 - 23:48

score 209 · Answer 4 · 2026-03-26 02:55:39

209

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Representing $\prod_{k=2}^{\infty}\left(1+\frac{1}{k^{2}-1}\right)$ as an Euler's product.

Published on 26 Mar 2026 - 2:55

score 62 · Answer 5 · 2013-04-29 19:54:16

62

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Prove the following: Product of Roots

Published on 29 Apr 2013 - 19:54

score 664 · Answer 6 · 2013-05-03 09:15:00

664

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sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$

Published on 03 May 2013 - 9:15

score 597 · Answer 7 · 2013-05-04 14:48:10

597

Views

The infinite product of the sines of all positive integers is zero

Published on 04 May 2013 - 14:48

score 270 · Answer 8 · 2013-05-12 03:38:13

270

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Infinite product involving powers of 2

Published on 12 May 2013 - 3:38

score 1.4k · Answer 9 · 2013-05-12 18:19:21

1.4k

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Closed form of $\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$

Published on 12 May 2013 - 18:19

score 919 · Answer 10 · 2026-03-29 05:50:10

919

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Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$

Published on 29 Mar 2026 - 5:50

score 188 · Answer 11 · 2013-05-17 02:39:57

188

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product of zeta(k)

Published on 17 May 2013 - 2:39

score 981 · Answer 12 · 2013-05-18 07:15:38

981

Views

Infinite products of a (finite) group

Published on 18 May 2013 - 7:15

score 438 · Answer 13 · 2013-05-23 13:15:33

438

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The limit of the product $\prod_{i=1}^n\frac{1 - (2i + 1)a/(2n)}{1 - ia/n}$ as $n\to\infty$

Published on 23 May 2013 - 13:15

score 122 · Answer 14 · 2013-05-28 15:44:01

122

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If $a_n\to+\infty$, $b_n\to0$, how about $\prod_{k=1}^n(1+b_ne^{-a_k})$?

Published on 28 May 2013 - 15:44

score 1.3k · Answer 15 · 2013-05-31 16:42:48

1.3k

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Proof of the product formula for sine function

Published on 31 May 2013 - 16:42

List Question

How to verify this equation $\prod_{j=1}^\infty{1+\exp(i2^{-j}\omega)\over2}={1-\exp(i\omega)\over i\omega}$?

Infinite Product $\prod_{n=1}^\infty\left(1+\frac1{\pi^2n^2}\right)$

Convergence of $\prod_{n=1}^\infty(1+a_n)$

Representing $\prod_{k=2}^{\infty}\left(1+\frac{1}{k^{2}-1}\right)$ as an Euler's product.

Prove the following: Product of Roots

sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$

The infinite product of the sines of all positive integers is zero

Infinite product involving powers of 2

Closed form of $\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$

Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$

product of zeta(k)

Infinite products of a (finite) group

The limit of the product $\prod_{i=1}^n\frac{1 - (2i + 1)a/(2n)}{1 - ia/n}$ as $n\to\infty$

If $a_n\to+\infty$, $b_n\to0$, how about $\prod_{k=1}^n(1+b_ne^{-a_k})$?

Proof of the product formula for sine function

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