The main object of the paper is to present a unified boundary integral method for numerical conformal mappings of bounded (see [M. M. S. Nasser
, Comput. Methods Funct. Theory 9, No. 1, 127–143 (2009; Zbl 1159.30007
)]) and unbounded (see, e.g., [S. Bergman
, The kernel function and conformal mapping, Mathematical Surveys. 5. Providence, R.I.: American Mathematical Society (AMS) (1950; Zbl 0040.19001
) (1970; Zbl 0208.34302
) (1980; Zbl 0473.30006
)]) multiply connected regions onto the five classical canonical slit domains, Using this method, the approximate parameters and mapping functions onto the five canonical slit domains can be computed by solving a linear systems with a common coefficient matrix. Several numerical examples are also given to support the effectiveness of the method.