These functions are here evaluated for complex orders and arguments via rapidly convergent series in the upper incomplete gamma function. The following machine-readable Derive expressions with corrections of some misprints and misconceptions should facilitate their implementation in computer-algebra systems. The auxiliary functions S(r) := r/(r^2)^(1/2) and sinc(x) := sin(x)/x are used.
1. "Bernoulli series" formula for the Riemann zeta function (eq. 15):
The summation limits are preset for about 10 decimal digits accuracy. This expansion beomes impractical for large Im(s) since the terms grow exponentially with Im(s) whereas zeta(s) grows only logarithmically. Note that the Bernoulli numbers BERNOULLI(n) vanish for odd n > 1.
2. "Riemann splitting" formula for the Riemann zeta function (eq. 16, 17):
Here, pi^s has been corrected to pi^(s/2). For 0 < Re(a) <= 1 if Im(a) <= 0, and for 0 <= Re(a) < 1 if Im(a) > 0, this formula returns the customary Hurwitz zeta function. Application of the functional equation zeta(s,a) = zeta(s,a+1) + a^(-s) then allows to compute zeta(s,a) for any other value of a. The Riemann zeta function is recovered for a=1: zeta(s,1) = zeta(s).
5. "Bernoulli series" formula for the polylogarithm (eq. 34):
Here, the omission of a prefactor (2 pi)^(s-1) has been corrected, and the polylogarithm argument has been converted from exp(-x) to z. This expansion holds for any z /= 1, for z=1 one has to substitute Li(s,1) = zeta(s). Note that Li(s,z) has a pole at z=1 if Re(s) <= 1.
6. "Riemann splitting" formula for the polylogarithm (eq. 37):
Here, the factors z^n and exp(- 2 pi i a u) have been corrected to 1. This expansion hold for all complex z, although it returns Li(s,1) = zeta(s) for Re(s) <= 1 too. Note that 2 cosh(n ln(z)) = z^n + z^(-n) and 2 sinh(n ln(z)) = z^n - z^(-n).
7. "Riemann splitting" formula for the Lerch transcendent (eq. 14):
For 0 < Re(a) <= 1 if Im(a) <= 0, and for 0 <= Re(a) < 1 if Im(a) > 0, this formula returns the customary Lerch transcendent (not some variant). Application of the functional equation Phi(z,s,a) = z Phi(z,s,a+1) + a^(-s) then allows to compute Phi(z,s,a) for any other value of a. The polylogarithm is recovered for a=1: z Phi(z,s,1) = Li(s,z). The expansion returns Phi(1,s,a) = zeta(s,a) for Re(s) <= 1 too, whereas Phi(z,s,a) has a pole at z=1 if Re(s) <= 1. Note that exp(- (2 pi i u + ln(z)) a) = exp(- 2 pi i u a) / z^a. The functional equation (7) for the Lerch transcendent is not entirely correct, the corrected equation was posted earlier this year in the thread "Inversion Lerch Phi".
Because the Derive kernel doesn't implement the upper incomplete gamma function, I had to define it in order to check the above expansions:
gamma_fraction1(s,z,m:=20):=GAMMA(s)-z^s*EXP(-z)/(ITERATE([s-z+i~ _*z/(q_+i_),i_-1],[q_,i_],[s,m],m)) SUB 1
gamma_fraction2(s,z,m:=20):=z^s*EXP(-z)/(ITERATE([z-(s-i_)*q_/(q~ _+i_),i_-1],[q_,i_],[z,m],m)) SUB 1
incomplete_gamma(s,z,m:=30):=IF([23*RE(z),6.5*IM(z)]^2<m^2 OR (R~ E(z)<0 AND 6.5*ABS(IM(z))<m),gamma_fraction1(s,z,m),gamma_fracti~ on2(s,z,m))
This delivers 10 decimal digits for arbitrary complex z, but only as long as s stays in the complex vicinity of s = 1/2. So this simple code is in need of refinement: Nothing is done about the cancellation of diverging terms near nonpositive integer s, and the fixed evaluation depth of the continued fractions produces inaccurate results if z happens to be close to a pole of the m-th convergent: the pole loci form chains that sweep across the z-plane as s and m vary, whereas my switching between the continued fractions is ignorant of s.
The series expansion of Gamma(s,z) in terms of generalized Laguerre polynomials (eq. 25) simply reproduces the convergents of Legendre's continued fraction (eq. 24):
gamma_series4(s,z,m:=20,ll):=PROG(ll:=VECTOR(MY_GENERALIZED_LAGU~ ERRE(n_,-s,-z),n_,0,m),z^s*EXP(-z)*SUM(PRODUCT(p_-s,p_,1,n_-1)/(~ n_!*ll SUB n_*ll SUB (n_+1)),n_,1,m))
(My redefinition of GENERALIZED_LAGUERRE(n,a,x) removes a minor problem with the implementation of the generalized Laguerre polynomials in Derive.) Thus, the series can be used to analyze the convergence behavior of the continued fraction, but it makes no sense at all to compute Gamma(s,z) in this way.
Convergents of the complementary continued fraction obey a very similar relation: