Stirling's Theorem - A Convergent Version of Stirling's Formula

A Convergent Version of Stirling's Formula

Thomas Bayes showed, in a letter to John Canton published by the Royal Society in 1763, that Stirling's formula did not give a convergent series.

Obtaining a convergent version of Stirling's formula entails evaluating

One way to do this is by means of a convergent series of inverted rising exponentials. If

then

where

where s(n, k) denotes the Stirling numbers of the first kind. From this we obtain a version of Stirling's series

begin{align} ln(Gamma (z)) & = left( z-tfrac{1}{2}right) ln(z) -z + tfrac{1}{2}ln(2 pi) + frac{1}{12(z+1)} + frac{1}{12(z+1)(z+2)} + \ & qquad qquad + frac{59}{360(z+1)(z+2)(z+3)} + frac{29}{60(z+1)(z+2)(z+3)(z+4)} + cdots end{align}

which converges when Re(z) > 0.

Read more about this topic:  Stirling's Theorem

Famous quotes containing the words formula, stirling and/or version:

    “My formula for greatness in human beings is amor fati: that one wants to change nothing, neither forwards, nor backwards, nor in all eternity. Not merely to endure necessity, still less to hide it—all idealism is mendacity in the face of necessity—but rather to love it.”
    —Friedrich Nietzsche (1844–1900)

    “Oh, if thy pride did not our joys control,
    What world of loving wonders shouldst thou see!
    For if I saw thee once transformed in me,
    Then in thy bosom I would pour my soul;”
    —William Alexander, Earl O Stirling (1580?–1640)

    “Truth cannot be defined or tested by agreement with ‘the world’; for not only do truths differ for different worlds but the nature of agreement between a world apart from it is notoriously nebulous. Rather—speaking loosely and without trying to answer either Pilate’s question or Tarski’s—a version is to be taken to be true when it offends no unyielding beliefs and none of its own precepts.”
    —Nelson Goodman (b. 1906)