nag_complex_log_gamma (s14agc) evaluates an approximation to the logarithm of the Gamma function 
defined for 
by
where 
is complex. It is extended to the rest of the complex plane by analytic continuation unless 
, in which case 
is real and each of the points 
is a singularity and a branch point.
nag_complex_log_gamma (s14agc) is based on the method proposed by Kölbig (1972) in which the value of 
is computed in the different regions of the 
plane by means of the formulae
![[[lnGAMMA(z),=,(z,-,1/2)lnz-z+1/2ln2Pi+z(∑)(B[2k])/(2k(2,k,-,1))z^(-2k)+R[K](z), x>=x[0]>=0 ],[,=,lnGAMMA(z,+,n)-ln(∏)(z,+,nu), x[0]>x>=0 ],[],[,=,lnPi-lnGAMMA(1,-,z)-ln(sinPiz), x<0 ]]](/support/helpjp/helpview.aspx?si=9747/file08640/math86.png)
where ![n = [x[0]]+[-x]](/support/helpjp/helpview.aspx?si=9747/file08640/math89.png)
, ![{B[2*k]}](/support/helpjp/helpview.aspx?si=9747/file08640/math91.png)
are Bernoulli numbers (see Abramowitz and Stegun (1972)) and ![[x]](/support/helpjp/helpview.aspx?si=9747/file08640/math93.png)
is the largest integer 
. Note that care is taken to ensure that the imaginary part is computed correctly, and not merely modulo 
.
The function uses the values 
and ![x[0] = 7](/support/helpjp/helpview.aspx?si=9747/file08640/math103.png)
. The remainder term ](/support/helpjp/helpview.aspx?si=9747/file08640/math105.png)
is discussed in Section [Accuracy].
To obtain the value of 
when 
is real and positive, s14abc (nag_log_gamma) can be used.
