Abstract analytic number theory

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.

Trigamma function ${\psi }_1(z)$ in the complex plane. The color of a point $z$ encodes the value of ${\psi }_1(z)$. Strong colors denote values close to zero and hue encodes the value's argument.

In mathematics, the trigamma function, denoted ${\psi }_1(z)$, is the second of the polygamma functions, and is defined by

${\psi }_1(z)=\frac{d^2}{d{z}^2}\ln \mathrm{\Gamma }(z)$.

It follows from this definition that

${\psi }_1(z)=\frac{d}{dz}\psi (z)$

where $\psi (z)$ is the digamma function. It may also be defined as the sum of the series

${\psi }_1(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{(z+n{)}^2},$

making it a special case of the Hurwitz zeta function

${\psi }_1(z)=\zeta (2,z).\frac{}{}$

Note that the last two formulæ are valid when $1-z$ is not a natural number.

Calculation

A double integral representation, as an alternative to the ones given above, may be derived from the series representation:

${\psi }_1(z)=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{y}{\int }}\frac{x^{z-1}y}{1-x}dxdy$

using the formula for the sum of a geometric series. Integration by parts yields:

${\psi }_1(z)=-\underset{0}{\overset{1}{\int }}\frac{x^{z-1}\ln x}{1-x}dx$

An asymptotic expansion as a Laurent series is

${\psi }_1(z)=\frac{1}{z}+\frac{1}{2z^2}+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{z^{2k+1}}=\sum _{k=0}^{\mathrm{\infty }}\frac{B_k}{z^{k+1}}$

if we have chosen $B_1=1/2$, i.e. the Bernoulli numbers of the second kind.

Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation

${\psi }_1(z+1)={\psi }_1(z)-\frac{1}{z^2}$

and the reflection formula

${\psi }_1(1-z)+{\psi }_1(z)=\frac{{\pi }^2}{{\sin }^2(\pi z)}$

which immediately gives the value for z=1/2.

Special values

The trigamma function has the following special values:

${\psi }_1\left(\frac{1}{4}\right)={\pi }^2+8K$

${\psi }_1\left(\frac{1}{2}\right)=\frac{{\pi }^2}{2}$

${\psi }_1(1)=\frac{{\pi }^2}{6}$

${\psi }_1\left(\frac{3}{2}\right)=\frac{{\pi }^2}{2}-4$

${\psi }_1(2)=\frac{{\pi }^2}{6}-1$

where K represents Catalan's constant.

There are no roots on the real axis of ${\psi }_1$, but there exist infinitely many pairs of roots $z_n,\overline{z_n}$ for $\mathrm{\Re }(z)<0$. Each such pair of root approach $\mathrm{\Re }(z_n)=-n+1/2$ quickly and their imaginary part increases slowly logarithmic with n. E.g. $z_1=-0.4121345\dots +i0.5978119\dots$ and $z_2=-1.4455692\dots +i0.6992608\dots$ are the first two roots with $\mathrm{\Im }(z)>0$.

Appearance

The trigamma function appears in the next surprising sum formula:

$\sum _{n=1}^{\mathrm{\infty }}\frac{n^2-\frac{1}{2}}{{(n^2+\frac{1}{2})}^2}[{\psi }_1(n-\frac{i}{\sqrt{2}})+{\psi }_1(n+\frac{i}{\sqrt{2}})]=-1+\frac{\sqrt{2}}{4}\pi \coth \left(\frac{\pi }{\sqrt{2}}\right)-\frac{3{\pi }^2}{4{\sinh }^2\left(\frac{\pi }{\sqrt{2}}\right)}+\frac{{\pi }^4}{12{\sinh }^4\left(\frac{\pi }{\sqrt{2}}\right)}(5+\cosh \left(\pi \sqrt{2}\right)).$

See also

• Gamma function
• Digamma function
• Polygamma function
• Catalan's constant

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. See section §6.4
• Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resource

æ