The Riemann zeta functions frequently occur in summation and integration problems.  The utility file ZetaFunctions.mth1CKZXC defines functions for approximating several functions related to the zeta function.  


ZETA(s) is the Riemann zeta function ζ(s).  If the real part of s is greater than 1, ζ(s) is defined as the sum of 1/k^s for k=1 to ∞.  Otherwise, it is defined by analytic continuation over the rest of the complex plane.  The function can be entered by typing ZETA on the expression entry line or by clicking on the ζ character on the math symbol toolbar.  ζ(s) simplifies to an exact closed form value if one is known.  For example,

VECTOR(ZETA(s), s, -4, 4)

simplifies to

⎡                                 2           4 ⎤
⎢     1           1      1       ð           ð  ⎥
⎢0, —————, 0, - ————, - ———, ∞, ————, æ(3), ————⎥
⎣    120         12      2        6          90 ⎦

If s is a real or complex number, ζ(s) approximates to a numerical value.


ZETA(s, z) is the Hurwitz or generalized zeta function ζ(s, z).  If the real part of s is greater than 1, ζ(s, z) is defined as the sum of 1/(k+z)^s for k=0 to ∞.  Otherwise, it is defined by analytic continuation over the rest of the complex plane.  Note that ζ(s, 1) is equal to ζ(s) for all s.  ζ(s, z) simplifies to an exact closed form value if one is known.  For example,

VECTOR(ZETA(s, 1/2), s, -4, 4)

simplifies to

⎡                              2             4 ⎤
⎢       7         1           ð             ð  ⎥
⎢0, - —————, 0, ————, 0, ±∞, ————, 7·æ(3), ————⎥
⎣      960       24            2             6 ⎦

A function for approximating ζ(s, z) is defined in the utility file ZetaFunctions.mth1CKZXC.


DILOG(z) is the dilogarithm function defined as the integral of LN(t)/(1-t) from t=1 to z.  For example, DILOG(1) is 0 and DILOG(0) is π²/6.  Antiderivatives and definite integrals involving logarithms having no closed-form in terms of the elementary functions often simplify to an expression involving DILOG.  For example,

INT(LN(x)/(x - 1), x)

simplifies to

DILOG(1 - x) + LN(x)·LN(1 - x)


Other Built-in Functions and ConstantsBuilt_in_Functions_and_Constants 

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