Conformal Moduli of Symmetric Circular Quadrilaterals With Cusps
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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