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Trigamma function
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<h2 id="calculation">Calculation</h2> <p>A <a href="/facts/Double_integral/Fvl3Bkam">double integral</a> representation, as an alternative to the ones given above, may be derived from the series representation: </p> ψ 1 ( z ) = ∫ 0 1 ∫ 0 x x z − 1 y ( 1 − x ) d y d x {\displaystyle \psi _{1}(z)=\int _{0}^{1}\!\!\int _{0}^{x}{\frac {x^{z-1}}{y(1-x)}}\,dy\,dx} <p>using the formula for the sum of a <a href="/facts/Geometric_series/Ne5TBH57">geometric series</a>. Integration over <i>y</i> yields: </p> ψ 1 ( z ) = − ∫ 0 1 x z − 1 ln x 1 − x d x {\displaystyle \psi _{1}(z)=-\int _{0}^{1}{\frac {x^{z-1}\ln {x}}{1-x}}\,dx} <p>An asymptotic expansion as a <a href="/facts/Laurent_series/sSMevFU8">Laurent series</a> can be obtained via the derivative of the <a href="/facts/Digamma_function/nDTNu86e">asymptotic expansion of the digamma function</a>: </p> ψ 1 ( z ) ∼ d d z ( ln z − ∑ n = 1 ∞ B n n z n ) = 1 z + ∑ n = 1 ∞ B n z n + 1 = ∑ n = 0 ∞ B n z n + 1 = 1 z + 1 2 z 2 + 1 6 z 3 − 1 30 z 5 + 1 42 z 7 − 1 30 z 9 + 5 66 z 11 − 691 2730 z 13 + 7 6 z 15 ⋯ {\displaystyle {\begin{aligned}\psi _{1}(z)&\sim {\operatorname {d} \over \operatorname {d} \!z}\left(\ln z-\sum _{n=1}^{\infty }{\frac {B_{n}}{nz^{n}}}\right)\\&={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {B_{n}}{z^{n+1}}}=\sum _{n=0}^{\infty }{\frac {B_{n}}{z^{n+1}}}\\&={\frac {1}{z}}+{\frac {1}{2z^{2}}}+{\frac {1}{6z^{3}}}-{\frac {1}{30z^{5}}}+{\frac {1}{42z^{7}}}-{\frac {1}{30z^{9}}}+{\frac {5}{66z^{11}}}-{\frac {691}{2730z^{13}}}+{\frac {7}{6z^{15}}}\cdots \end{aligned}}} <p>where <i>B</i><i>n</i> is the <i>n</i>th <a href="/facts/Bernoulli_number/RH0mblhs">Bernoulli number</a> and we choose <i>B</i>1 = 1/2. </p> <h3>Recurrence and reflection formulae</h3> <p>The trigamma function satisfies the <a href="/facts/Recurrence_relation/JolXC71M">recurrence relation</a> </p> ψ 1 ( z + 1 ) = ψ 1 ( z ) − 1 z 2 {\displaystyle \psi _{1}(z+1)=\psi _{1}(z)-{\frac {1}{z^{2}}}} <p>and the <a href="/facts/Reflection_formula/aKUhTbH0">reflection formula</a> </p> ψ 1 ( 1 − z ) + ψ 1 ( z ) = π 2 sin 2 π z {\displaystyle \psi _{1}(1-z)+\psi _{1}(z)={\frac {\pi ^{2}}{\sin ^{2}\pi z}}\,} <p>which immediately gives the value for <i>z</i> = 1/2: ψ 1 ( 1 2 ) = π 2 2 {\displaystyle \psi _{1}({\tfrac {1}{2}})={\tfrac {\pi ^{2}}{2}}} . </p> <h3>Special values</h3> <p>At positive integer values we have that </p> ψ 1 ( n ) = π 2 6 − ∑ k = 1 n − 1 1 k 2 , ψ 1 ( 1 ) = π 2 6 , ψ 1 ( 2 ) = π 2 6 − 1 , ψ 1 ( 3 ) = π 2 6 − 5 4 . {\displaystyle \psi _{1}(n)={\frac {\pi ^{2}}{6}}-\sum _{k=1}^{n-1}{\frac {1}{k^{2}}},\qquad \psi _{1}(1)={\frac {\pi ^{2}}{6}},\qquad \psi _{1}(2)={\frac {\pi ^{2}}{6}}-1,\qquad \psi _{1}(3)={\frac {\pi ^{2}}{6}}-{\frac {5}{4}}.} <p> At positive half integer values we have that </p> ψ 1 ( n + 1 2 ) = π 2 2 − 4 ∑ k = 1 n 1 ( 2 k − 1 ) 2 , ψ 1 ( 1 2 ) = π 2 2 , ψ 1 ( 3 2 ) = π 2 2 − 4. {\displaystyle \psi _{1}\left(n+{\frac {1}{2}}\right)={\frac {\pi ^{2}}{2}}-4\sum _{k=1}^{n}{\frac {1}{(2k-1)^{2}}},\qquad \psi _{1}\left({\tfrac {1}{2}}\right)={\frac {\pi ^{2}}{2}},\qquad \psi _{1}\left({\tfrac {3}{2}}\right)={\frac {\pi ^{2}}{2}}-4.} <p>The trigamma function has other special values such as: </p> ψ 1 ( 1 4 ) = π 2 + 8 G {\displaystyle \psi _{1}\left({\tfrac {1}{4}}\right)=\pi ^{2}+8G} <p>where G represents <a href="/facts/Catalan%27s_constant/c9fNqG8j">Catalan's constant</a>. </p><p>There are no roots on the real axis of <i>ψ</i>1, but there exist infinitely many pairs of roots <i>zn</i>, <i>zn</i> for Re <i>z</i> < 0. Each such pair of roots approaches Re <i>zn</i> = −<i>n</i> + 1/2 quickly and their imaginary part increases slowly logarithmic with n. For example, <i>z</i>1 = −0.4121345... + 0.5978119...<i>i</i> and <i>z</i>2 = −1.4455692... + 0.6992608...<i>i</i> are the first two roots with Im(<i>z</i>) > 0. </p> <h3>Relation to the Clausen function</h3> <p>The <a href="/facts/Digamma_function/nDTNu86e">digamma function</a> at rational arguments can be expressed in terms of trigonometric functions and logarithm by the <a href="/facts/Digamma_function/nDTNu86e">digamma theorem</a>. A similar result holds for the trigamma function but the circular functions are replaced by <a href="/facts/Clausen_function/1mDkvwcL">Clausen's function</a>. Namely,<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> ψ 1 ( p q ) = π 2 2 sin 2 ( π p / q ) + 2 q ∑ m = 1 ( q − 1 ) / 2 sin ( 2 π m p q ) Cl 2 ( 2 π m q ) . {\displaystyle \psi _{1}\left({\frac {p}{q}}\right)={\frac {\pi ^{2}}{2\sin ^{2}(\pi p/q)}}+2q\sum _{m=1}^{(q-1)/2}\sin \left({\frac {2\pi mp}{q}}\right){\textrm {Cl}}_{2}\left({\frac {2\pi m}{q}}\right).} <h2 id="appearance">Appearance</h2> <p>The trigamma function appears in this sum formula:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> ∑ n = 1 ∞ n 2 − 1 2 ( n 2 + 1 2 ) 2 ( ψ 1 ( n − i 2 ) + ψ 1 ( n + i 2 ) ) = − 1 + 2 4 π coth π 2 − 3 π 2 4 sinh 2 π 2 + π 4 12 sinh 4 π 2 ( 5 + cosh π 2 ) . {\displaystyle \sum _{n=1}^{\infty }{\frac {n^{2}-{\frac {1}{2}}}{\left(n^{2}+{\frac {1}{2}}\right)^{2}}}\left(\psi _{1}{\bigg (}n-{\frac {i}{\sqrt {2}}}{\bigg )}+\psi _{1}{\bigg (}n+{\frac {i}{\sqrt {2}}}{\bigg )}\right)=-1+{\frac {\sqrt {2}}{4}}\pi \coth {\frac {\pi }{\sqrt {2}}}-{\frac {3\pi ^{2}}{4\sinh ^{2}{\frac {\pi }{\sqrt {2}}}}}+{\frac {\pi ^{4}}{12\sinh ^{4}{\frac {\pi }{\sqrt {2}}}}}\left(5+\cosh \pi {\sqrt {2}}\right).} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Gamma_function/SjGQcnyw">Gamma function</a></li> <li><a href="/facts/Digamma_function/nDTNu86e">Digamma function</a></li> <li><a href="/facts/Polygamma_function/IF6W68aL">Polygamma function</a></li> <li><a href="/facts/Catalan%27s_constant/c9fNqG8j">Catalan's constant</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Milton Abramowitz and Irene A. Stegun, <i><a href="/facts/Abramowitz_and_Stegun/CNGSV3Ed">Handbook of Mathematical Functions</a></i>, (1964) Dover Publications, New York. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-61272-4. See section <a href="http://www.math.sfu.ca/~cbm/aands/page_260.htm">§6.4</a></li> <li>Eric W. Weisstein. <a href="http://mathworld.wolfram.com/TrigammaFunction.html">Trigamma Function -- from MathWorld--A Wolfram Web Resource</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lewin, L., ed. (1991). Structural properties of polylogarithms. American Mathematical Society. ISBN 978-0821816349. <a href="978-0821816349" target="_blank">978-0821816349</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Mező, István (2013). "Some infinite sums arising from the Weierstrass Product Theorem". Applied Mathematics and Computation. 219 (18): 9838–9846. doi:10.1016/j.amc.2013.03.122. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>