Parametric dependence of irregular conformal block

Irregular conformal block is an important tool to study a new type of conformal theories, which can be constructed as the colliding limit of the regular conformal block. The irregular conformal block is realized as the $\beta$-deformed Penner matrix model whose partition function is regarded as the inner product of the irregular modules. The parameter dependence of the inner product is obtained explicitly using the loop equation with close attention to singularities in the parameter space. It is noted that the exact singular structure of the parameter space in general can be found using a very simple and powerful method which uses the flow equations of the partition function together with the hierarchical structure of the singularity. This method gives the exact expression to all orders of large $N$ expansion without using the explicit contour integral of the filling fraction.