The psi function, also known as the digamma function, is a special function in mathematics that represents the logarithmic derivative of the gamma function. It is denoted as ψ(x) and is defined as the derivative of the natural logarithm of the gamma function, which is a generalization of factorials to non-integer values. The psi function is useful in various areas of mathematics, including number theory and statistics. The psi function has several important properties, such as being related to the harmonic numbers and having specific values at integer points. It is often used in calculus and complex analysis, where it helps in solving problems involving series and integrals. The function can also be extended to complex numbers, making it valuable in advanced mathematical applications.