The Dirichlet Beta function:

Beta is defined for all values of x.

For example, Beta(1) = 1 − 1/3 + 1/5 − 1/7 + ... = Pi/4 = 7.85398,16339,74483,09615,66...E−1.

Here are some notes from my program XPCalc - Extra Precision Floating-Point Calculator http://www.oocities.org/hjsmithh/download.html#XPCalc :

Beta(x) = Dirichlet Beta function:

Dirichlet Beta function of x > 0 is defined by the infinite series 1 − 1/3^x + 1/5^x − 1/7^x + ... .

For x >= 0.5, Beta(x) is computed by

with the k-th appended term being x*(x+1)*...*(x+k−2)*2^k * (2^k − 1) * B(k) / (2*k!*(J^(k+x−1))). B(k) is a Bernoulli number and J is a large number of the form 4n + 1. This is from "An Atlas Of Functions" by Spanier, J. and Oldham, K. B. 1987, equation 3:3:7.

For x < 0.5, the reflection formula is used: let y = 1−x, then

Beta(0) = 1/2. Beta(1) = Pi/4. Beta(2) = G = Catalan's constant = 0.915965594177219015054603514932384110774149374282... . Beta(x) = 0 for all negative odd integers.

BetaL(X) = Dirichlet beta function for large |X|:## z = Beta(x) // Dirichlet Beta function for x >= 0.5 { if (x == 0) return 1/2; if ((x < 0) && (x is odd)) return 0; sum = 0; // Beta = Sum ... n = (int)((decimal-digits desired / 300.0) * digits + 5.0); if (n < 300) n = 300; j = 4 * n + 1; sign = −1; // (−1)^(k−1) for (int i = 3; i < j; i += 2) { sum += sign * (i ^ −x); sign = −sign; } sum += (j ^ −x) / 2; twoToK = 4; d = 4 * (j ^ (x + 1)); // denominator 2*k!*(J^(k+x−1)) with k = 2 xK = x; // xK = x + k − 2 xKP = xK; // xKP = x * (x+1) * (x+2) * ... * (x+k−2) GenBernI(limBern); // generate Bernoulli numbers for (int k = 2; ; k += 2) { term = (twoToK − 1) * twoToK * xKP; xK++; xKP *= xK; if (k > 2) { term *= xK; xK++; xKP += xK; } term *= B(k) / d; // B(k) = k-th Bernoulli number sumN = sum + term; // sumN = new sum if (sumN == sum) break; sum = sumN; twoToK *= 4; // twoToK = 2^k d *= j * j * (k+1) * (k+2); } sum += 1; return sum; }

Same as Beta(x) but uses a method that is faster for large |x|. The method comes from the book "Mathematical Constants" by Steven Finch. On page 53 there is a formula for the Dirichlet beta function as a product of primes. See:

BetaL(x) = Prod{p prime, p = 1 mod 4}[p^x/(p^x − 1)] * Prod{ p prime, p = 3 mod 4}[ p^x/(p^x + 1)].

This product is repeated until a factor is equal to 1. If |x| < 100, this converges very slowly, so the normal Beta(x) is used.

For x < 0.5, the reflection formula is used: let y = 1−x, then

BetaL(x) = (2/Pi)^y * Sin(Pi*y/2) * Gam(y) * BetaL(y).

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