trigamma

The trigamma function, denoted by ψ(x), is a mathematical function that is closely related to the digamma function, which is itself related to the gamma function. It is defined as the derivative of the digamma function, and can be expressed as the logarithmic derivative of the gamma function.

Mathematically, the trigamma function can be defined as ψ(x) = d/dx (ψ(x)), where ψ(x) is the digamma function. This relationship is fundamental to the study of the trigamma function, and is used to derive its properties and asymptotic behavior.

The trigamma function has several important properties, including its asymptotic behavior as x approaches 0 and infinity, as well as its relation to the Riemann zeta function. Specifically, the trigamma function is closely tied to the asymptotic expansion of the gamma function, which is a crucial result in number theory and analysis.

The trigamma function has applications in various areas of mathematics, such as number theory, analysis, and combinatorics. In particular, it is used in the study of the properties of the gamma function, as well as in the asymptotic analysis of exponential sums and integrals. Additionally, the trigamma function plays a key role in the study of continued fractions and Hilbert transforms.

The trigamma function can be computed numerically using various methods, including the use of asymptotic approximations and Chebyshev expansion. These methods allow for efficient and accurate computation of the trigamma function for large values of x.

Overall, the trigamma function is a fundamental mathematical object that has far-reaching implications for the study of special functions and their applications in analysis, number theory, and algebra.