Abstract
We 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 \(\Omega \).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Shorey, T.N. (2020). Harmonic Functions. In: Complex Analysis with Applications to Number Theory. Infosys Science Foundation Series(). Springer, Singapore. https://doi.org/10.1007/978-981-15-9097-9_4
Download citation
DOI: https://doi.org/10.1007/978-981-15-9097-9_4
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-9096-2
Online ISBN: 978-981-15-9097-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)