Abstract
AbstractWe introduce the Cauchy–Riemann equations in Sect. 4.2 and the harmonic functions in Sect. 4.3 and show that real and imaginary parts of an analytic function are harmonic. We prove the existence of a harmonic conjugate of a harmonic function in a simply connected region in Sect. 4.4 where we also prove its converse. We introduce continuous functions with Mean Value Property in a region \(\Omega \) in Sect. 4.5 and prove in Sect. 4.5 Maximum principle for harmonic functions in a region \(\Omega \) satisfying MVP in \(\Omega \) and Sect. 4.6 that such functions characterise harmonic functions in \(\Omega \).
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