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| | [[Image:Complex Polygamma 1.jpg|right|thumb|300px|Trigamma function <math> \psi_1(z) </math> in the [[complex plane]]. The color of a point <math> z </math> encodes the value of <math> \psi_1(z) </math>. Strong colors denote values close to zero and hue encodes the value's [[complex number|argument]].]] |
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| | In [[mathematics]], the '''trigamma function''', denoted <math> \psi_1(z) </math>, is the second of the [[polygamma function]]s, and is defined by |
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| | using the formula for the sum of a [[geometric series]]. Integration by parts yields: |
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| == petrochemicals Tiffany Jewellery Australia Brisbane == | | : <math> \psi_1(z + 1) = \psi_1(z) - \frac{1}{z^2}</math> |
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| == 7/29/13 Longchamp Sydney == | | : <math> \psi_1(1 - z) + \psi_1(z) = \frac{\pi^2}{\sin^2(\pi z)} \,</math> |
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| == Greenspan Louis Vuitton Australia == | | ===Special values=== |
| | The trigamma function has the following special values: |
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| | : <math> \psi_1\left(\frac{3}{2}\right) = \frac{\pi^2}{2} - 4</math> |
|
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| | : <math> \psi_1(2) = \frac{\pi^2}{6} - 1</math> |
|
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| | where K represents [[Catalan's constant]]. |
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| | There are no roots on the real axis of <math>\psi_1</math>, but there exist infinitely many pairs of roots <math>z_n, \overline{z_n}</math> for <math>\Re(z) < 0</math>. Each such pair of root approach <math>\Re(z_n)= -n+1/2</math> quickly and their imaginary part increases slowly logarithmic with n. |
|
| | E.g. <math>z_1 = -0.4121345\ldots + i 0.5978119\ldots</math> and <math>z_2 = -1.4455692\ldots + i 0.6992608\ldots</math> are the first two roots with <math>\Im(z) > 0</math>. |
| </ul>
| | |
| | ==Appearance== |
| | |
| | The trigamma function appears in the next surprising sum formula:<ref name=mezo/> |
| | |
| | : <math> \sum_{n=1}^\infty\frac{n^2-\frac12}{\left(n^2+\frac12\right)^2}\left[\psi_1\left(n-\frac{i}{\sqrt{2}}\right)+\psi_1\left(n+\frac{i}{\sqrt{2}}\right)\right]= |
| | -1+\frac{\sqrt{2}}{4}\pi\coth\left(\frac{\pi}{\sqrt{2}}\right)-\frac{3\pi^2}{4\sinh^2\left(\frac{\pi}{\sqrt{2}}\right)}+\frac{\pi^4}{12\sinh^4\left(\frac{\pi}{\sqrt{2}}\right)}\left(5+\cosh\left(\pi\sqrt{2}\right)\right). |
| | </math> |
| | |
| | ==See also== |
| | * [[Gamma function]] |
| | * [[Digamma function]] |
| | * [[Polygamma function]] |
| | * [[Catalan's constant]] |
| | |
| | ==Notes== |
| | {{reflist|refs= |
| | <ref name=mezo> |
| | {{Cite journal |
| | | last1 = Mező | first1 = István |
| | | title = Some infinite sums arising from the Weierstrass Product Theorem |
| | | journal = Applied Mathematics and Computation |
| | | volume = 219 |
| | | pages = 9838–9846 |
| | | year = 2013 |
| | | url = http://www.sciencedirect.com/science/article/pii/S0096300313003834 |
| | | doi=10.1016/j.amc.2013.03.122 |
| | }}</ref> |
| | }} |
| | |
| | ==References== |
| | * Milton Abramowitz and Irene A. Stegun, ''[[Abramowitz and Stegun|Handbook of Mathematical Functions]]'', (1964) Dover Publications, New York. ISBN 0-486-61272-4. See section [http://www.math.sfu.ca/~cbm/aands/page_260.htm §6.4] |
| | * Eric W. Weisstein. [http://mathworld.wolfram.com/TrigammaFunction.html Trigamma Function -- from MathWorld--A Wolfram Web Resource] |
| | |
| | æ |
| | |
| | [[Category:Gamma and related functions]] |
| | |
| | [[hu:Trigamma-függvény]] |