  By using Cauchy integral formula in the theory of complex functions, the
authors establish some integral representations for the principal branches of
several complex functions involving the logarithmic function, find some
properties, such as being operator monotone function, being complete Bernstein
function, and being Stieltjes function, for these functions, and verify a
conjecture on complete monotonicity of a function involving the logarithmic
function.

Classical Analysis and ODEs