Conformal Moduli of Symmetric Circular Quadrilaterals With Cusps
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We investigate moduli of planar circular quadrilaterals symmetric with respect to both the coordinate axes. First, making use the Schwarz ODE, we develop an analytic method to determine a conformal mapping the unit disk onto a given circular quadrilateral. We devise a numerical method to find the accessory parameters, solve the equation and to compute the modulus of the quadrilateral. We also give an example of a circular quadrilateral for which the value of the conformal modulus can be found in the analytic form; this example is used to validate the numeric calculations. We also use another method, so called hp-FEM, for the numeric calculation of the moduli. These two different approaches provide results agreeing with high accuracy. An example of a quadrilateral, connected with a circular hexagon, with known modulus is considered as well.
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