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I need to integrate the following integral:

\begin{align} I = \int^z\frac{1-\zeta^2}{(1+\zeta^2)(\zeta-\zeta_l)(1-\zeta_l\zeta)}\prod_{k=2}^{n-1}\left ( \frac{\zeta-z_k}{1-\zeta z_k} \right )^{-\beta_k} d\zeta \end{align}

along any complex path within the upper half of the unit circle in the argand plane, where $z_k$, $\beta_k$ and $\zeta_l$ have prescribed values.

I am wondering if anyone can suggest a way to robustly calculate this integral with their own code, or comment on if what I am doing below is sensible:

The integral arises in Kirchoff flow problems and is part of the Schwarz Christoffel Parameter problem. I have modified the following code from Toby Driscoll in an attempt to compute my integral. Note that this assumes that qdat which is a matrix of the Gauss Jacobi Quadrature weights is available.

As explained in the comments below, the idea is to refine the interval on which quadrature is applied such that any singularities $z_k$ (other than the end point of integration, are farther away than at least half the integration interval. This is coined "adaptive or compound" Gauss Jacobi Quadrature. Because of the form of the integrand (essentially it is of the type $\int f(x) (1+x)^\alpha(1-x)^\beta dx$) it is amenable to Gauss Jacobi Quadrature.

function I = modified_hpquad(z1,z2,sing1,z,beta,qdat)
%HPQUAD (not intended for calling directly by the user)
%   Numerical quadrature for the half-plane map.

%   Copyright 1998 by Toby Driscoll.
%   $Id: hpquad.m 298 2009-09-15 14:36:37Z driscoll $

%   HPQUAD(z1,z2,sing1,z,beta,qdat)
%   z1,z2 are vectors of left and right endpoints.  sing1 is a vector of
%   integer indices which label the singularities in z1.  So if sing1(5)
%   = 3, then z1(5) = z(3).  A zero means no singularity.  z is the
%   vector of finite singularities; beta is the vector of associated
%   turning angles.  qdat is quadrature data from SCQDATA.
%
%   Make sure z and beta are column vectors.
%   
%   The integral is subdivided, if necessary, so that no singularity
%   lies closer to the left endpoint than 1/2 the length of the
%   integration (sub)interval.


%Some parameters for modified SCT:
sl = 0;


nqpts = size(qdat,1);
% Note: Here n is the total number of *finite* singularities; i.e., the
% number of terms in the product appearing in the integrand.
n = length(z);

bigz = z(:,ones(1,nqpts));
bigbeta = beta(:,ones(1,nqpts));
if isempty(sing1)
  sing1 = zeros(length(z1),1);
end

%I = zeros(size(z1));
I = 0;
%nontriv = find(z1(:)~=z2(:))';
nontriv = find(z1~=z2);

if nontriv == 1

    za = z1; 
    zb = z2;
    sng = sing1;

    if sng > n
        sng = 0; %No singularity 
    end

    % Allowable integration step, based on nearest singularity.
    dist = min(1,2*min(abs(z([1:sng-1,sng+1:n])-za))/abs(zb-za)); % z([1:sng-1,sng+1:n]) jumps over sing
    zr = za + dist*(zb-za);
    ind = rem(sng+n,n+1)+1;

    % Adjust Gauss-Jacobi nodes and weights to interval.
    nd = ((zr-za)*qdat(:,ind) + zr + za).'/2; % G-J nodes
    wt = ((zr-za)/2) * qdat(:,ind+n+1);     % G-J weights
    %terms = nd(ones(n,1),:) - bigz;
    terms = (nd(ones(n,1),:) - bigz)./(1-bigz.*nd(ones(n,1),:)); %Modified SC
    if any(terms(:)==0) 
        % Endpoints are practically coincident.
        I = 0;
    else
        % Use Gauss-Jacobi on first subinterval, if necessary.
        if sng > 0
        terms(sng,:) = terms(sng,:)./abs(terms(sng,:));
        wt = wt*(abs(zr-za)/2)^beta(sng);
        end
        %I(k) = exp(sum(log(terms).*bigbeta,1))*wt;
        I = exp(log((1-nd.^2)./((1+nd.^2).*(nd-sl).*(1-sl.*nd)))+sum(log(terms).*bigbeta,1))*wt;
        while dist < 1              
            % Do regular Gaussian quad on other subintervals.
            zl = zr;
            dist = min(1,2*min(abs(z-zl))/abs(zl-zb));
            zr = zl + dist*(zb-zl);
            nd = ((zr-zl)*qdat(:,n+1) + zr + zl).'/2;
            wt = ((zr-zl)/2) * qdat(:,2*n+2);
            %terms = nd(ones(n,1),:) - bigz;
            terms = (nd(ones(n,1),:) - bigz)./(1-bigz.*nd(ones(n,1),:));
            I = I + exp(log((1-nd.^2)./((1+nd.^2).*(nd-sl).*(1-sl.*nd)))+sum(log(terms).*bigbeta,1))*wt;
        end
    end
end

The code above is a modified version of a code which does exactly what I want for the integral

\begin{align} I = \int^z\prod_{k=1}^{n-1}\left ( \zeta-z_k \right )^{-\beta_k} d\zeta \end{align}

and I have copied that code down for reference here:

function I = hpquad(z1,z2,varargin)
%HPQUAD (not intended for calling directly by the user)
%   Numerical quadrature for the half-plane map.

%   Copyright 1998 by Toby Driscoll.
%   $Id: hpquad.m 298 2009-09-15 14:36:37Z driscoll $

%   HPQUAD(z1,z2,sing1,z,beta,qdat)
%   z1,z2 are vectors of left and right endpoints.  sing1 is a vector of
%   integer indices which label the singularities in z1.  So if sing1(5)
%   = 3, then z1(5) = z(3).  A zero means no singularity.  z is the
%   vector of finite singularities; beta is the vector of associated
%   turning angles.  qdat is quadrature data from SCQDATA.
%
%   Make sure z and beta are column vectors.
%   
%   HPQUAD integrates from a possible singularity at the left end to a
%   regular point at the right.  If both endpoints are singularities,
%   you must break the integral into two pieces and make two calls, or
%   call HPQUAD(z1,z2,sing1,sing2,z,beta,qdat) and accept an automatic
%   choice. 
%   
%   The integral is subdivided, if necessary, so that no singularity
%   lies closer to the left endpoint than 1/2 the length of the
%   integration (sub)interval.

if nargin==7
  % Break into two pieces with recursive call.
  [sing1,sing2,z,beta,qdat] = deal(varargin{:});
  mid = (z1+z2)/2;
  mid = mid + 1i*abs(mid);
  I1 = hpquad(z1,mid,sing1,z,beta,qdat);
  I2 = hpquad(z2,mid,sing2,z,beta,qdat);
  I = I1-I2;
  return
else
  [sing1,z,beta,qdat] = deal(varargin{:});
end




nqpts = size(qdat,1);
% Note: Here n is the total number of *finite* singularities; i.e., the
% number of terms in the product appearing in the integrand.
n = length(z);
bigz = z(:,ones(1,nqpts));
bigbeta = beta(:,ones(1,nqpts));
if isempty(sing1)
  sing1 = zeros(length(z1),1);
end

I = zeros(size(z1));
nontriv = find(z1(:)~=z2(:))';



for k = nontriv
  za = z1(k);
  zb = z2(k);
  sng = sing1(k);

  % Allowable integration step, based on nearest singularity.
  dist = min(1,2*min(abs(z([1:sng-1,sng+1:n])-za))/abs(zb-za));
%%  if isempty(dist), dist=1; end
  zr = za + dist*(zb-za);
  ind = rem(sng+n,n+1)+1;
  % Adjust Gauss-Jacobi nodes and weights to interval.
  nd = ((zr-za)*qdat(:,ind) + zr + za).'/2; % G-J nodes
  wt = ((zr-za)/2) * qdat(:,ind+n+1);   % G-J weights
  terms = nd(ones(n,1),:) - bigz;
  if any(terms(:)==0) 
    % Endpoints are practically coincident.
    I(k) = 0;
  else
    % Use Gauss-Jacobi on first subinterval, if necessary.
    if sng > 0
      terms(sng,:) = terms(sng,:)./abs(terms(sng,:));
      wt = wt*(abs(zr-za)/2)^beta(sng);
    end
    I(k) = exp(sum(log(terms).*bigbeta,1))*wt;
    while dist < 1              
      % Do regular Gaussian quad on other subintervals.
      zl = zr;
      dist = min(1,2*min(abs(z-zl))/abs(zl-zb));
      zr = zl + dist*(zb-zl);
      nd = ((zr-zl)*qdat(:,n+1) + zr + zl).'/2;
      wt = ((zr-zl)/2) * qdat(:,2*n+2);
      terms = nd(ones(n,1),:) - bigz;
      I(k) = I(k) + exp(sum(log(terms).*bigbeta,1)) * wt;
    end
  end
end
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