We compute the Brown measure of the sum of a self-adjoint element and an elliptic element. We prove that the push-forward of this Brown measure of a natural map is the law of the free convolution of the self-adjoint element and the semicircle law; it is also a push-forward measure of the Brown measure of the sum of the self-adjoint element and a circular element by another natural map. We also study various asymptotic behaviors of this family of Brown measures as the variance of the elliptic element approaches infinity.
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