In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function (it is an instance of a pseudogamma function). This function, with its argument shifted down by 1, interpolates the factorial and extends it to real and complex numbers in a different way than Euler's gamma function. It is defined as:
where Γ(x) denotes the classical gamma function. If n is a positive integer, then:
Unlike the classical gamma function, Hadamard's gamma function H(x) is an entire function, i.e. it has no poles in its domain. It satisfies the functional equation
with the understanding that is taken to be 0 for positive integer values of x.
Hadamard's gamma can also be expressed as
where is the Lerch zeta function, and as
where ψ(x) denotes the digamma function.
In mathematics, the gamma function is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function is defined for all complex numbers except non-positive integers, and for every positive integer , The gamma function can be defined via a convergent improper integral for complex numbers with positive real part:
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as for , and its analytic continuation elsewhere.
Euler's constant is a mathematical constant, usually denoted by the lowercase Greek letter gamma, defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function:
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, … by
In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function:
In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted A, is a mathematical constant, related to special functions like the K-function and the Barnes G-function. The constant also appears in a number of sums and integrals, especially those involving the gamma function and the Riemann zeta function. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin.
In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.
In mathematics, the Dirichlet beta function is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four.
In mathematics, the reciprocal gamma function is the function
In mathematics, a reflection formula or reflection relation for a function f is a relationship between f(a − x) and f(x). It is a special case of a functional equation. It is common in mathematical literature to use the term "functional equation" for what are specifically reflection formulae.
Carl Johan Malmsten was a Swedish mathematician and politician. He is notable for early research into the theory of functions of a complex variable, for the evaluation of several important logarithmic integrals and series, for his studies in the theory of Zeta-function related series and integrals, as well as for helping Mittag-Leffler start the journal Acta Mathematica. Malmsten became Docent in 1840, and then, Professor of mathematics at the Uppsala University in 1842. He was elected a member of the Royal Swedish Academy of Sciences in 1844. He was also a minister without portfolio in 1859–1866 and Governor of Skaraborg County in 1866–1879.
In discrete calculus the indefinite sum operator, denoted by or , is the linear operator, inverse of the forward difference operator . It relates to the forward difference operator as the indefinite integral relates to the derivative. Thus
In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll.
In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution.
The Bernoulli polynomials of the second kindψn(x), also known as the Fontana–Bessel polynomials, are the polynomials defined by the following generating function:
In mathematics, a pseudogamma function is a function that interpolates the factorial. The gamma function is the most famous solution to the problem of extending the notion of the factorial beyond the positive integers only. However, it is clearly not the only solution, as, for any set of points, an infinite number of curves can be drawn through those points. Such a curve, namely one which interpolates the factorial but is not equal to the gamma function, is known as a pseudogamma function. The two most famous pseudogamma functions are Hadamard's gamma function:
