> restart:
> mu:=alpha*GAMMA(1+1/beta);
> var:=alpha^2*(GAMMA(1+2/beta)-GAMMA(1+1/beta)^2);
> V:=alpha^2*(GAMMA(1+1/beta)-GAMMA(1+1/beta)^2);

                                            1
                     mu := alpha GAMMA(1 + ----)
                                           beta


                      2 /           2                 1   2\
          var := alpha  |GAMMA(1 + ----) - GAMMA(1 + ----) |
                        \          beta              beta  /


                     2 /           1                 1   2\
           V := alpha  |GAMMA(1 + ----) - GAMMA(1 + ----) |
                       \          beta              beta  /

> with(plots):
> p1:=plot(V/var,beta=0...30,color='red'):
> expr1:=evalf(convert(series(V/var,beta=infinity,3),polynom)):
> p2:=plot(expr1,beta=0...30,color='blue'):
> x:=fsolve(diff(V/var,beta),beta):
> expr:=convert(series(V/var,beta=x,3),polynom):
> p3:=plot(expr,beta=0...2,color='green'):
> display({p1,p2,p3});

> expr1;

                          0.3509050462 beta

> solve({mu=m,k*beta*V=v},{alpha,beta});
Warning, solutions may have been lost

> M:=map(factor,<<diff(mu,alpha)|diff(mu,beta)>,<diff(var,alpha)|diff(va
> r,beta)>>);

  M :=

        [                              beta + 1        beta + 1 ]
        [                    alpha Psi(--------) GAMMA(--------)]
        [      beta + 1                  beta            beta   ]
        [GAMMA(--------) , - -----------------------------------]
        [        beta                           2               ]
        [                                   beta                ]

        [         /       beta + 2          beta + 1 2\          2 /
        [-2 alpha |-GAMMA(--------) + GAMMA(--------) | , 2 alpha  |
        [         \         beta              beta    /            \

             beta + 2        beta + 2
        -Psi(--------) GAMMA(--------)
               beta            beta

                 beta + 1 2     beta + 1 \   /     2]
         + GAMMA(--------)  Psi(--------)|  /  beta ]
                   beta           beta   / /        ]

> help(Psi);
> r:=(log(k)-Psi(k)-S)/(Psi(1,k)-1/k);

                            ln(k) - Psi(k) - S
                       r := ------------------
                             Psi(1, k) - 1/k

> p:=expand(evalf(convert(series(r,k),polynom)));

            2                      2         2                3
  p := k + k  ln(k) + 1.577215665 k  - 1. S k  - 1.712652469 k

            3               3                4                4
         + k  ln(k) - 1. S k  - 0.700897539 k  - 0.644934068 k  ln(k)

                          4                5                5
         + 0.644934068 S k  + 1.578815873 k  + 0.114245670 k  ln(k)

                          5
         - 0.114245670 S k

> series(solve(log(k)-Psi(k)-S,k),_Z);

                    RootOf(-ln(_Z) + Psi(_Z) + S)

> n:=1;evalf(series(solve(convert(series(log(k)-Psi(k)-S,k,n),polynom),k
> ),S));

                                n := 1


   - 0.03381041981 + 0.6131875243 I + (0.2506324443 + 0.1887666633 I)

                                              2
        S + (0.1410530934 - 0.01305786841 I) S  +

                                           3
        (0.04710451564 - 0.03604051692 I) S  +

                                           4
        (0.01169159185 - 0.02083727925 I) S  +

                                             5      6
        (0.002334197122 - 0.009323936491 I) S  + O(S )

> 
