7 Cauchy Riemann Equations and Examples #cauchyriemann

In this important lecture, we explore the Cauchy-Riemann equations, a key criterion for the differentiability of complex functions. We begin by deriving the Cauchy-Riemann equations and then apply them to various examples to understand where a function is not differentiable, despite appearances.

We also emphasize that satisfying the Cauchy-Riemann equations at a point is necessary but not sufficient for differentiability. This subtle yet crucial idea is reinforced through carefully chosen counterexamples.

🧠 Topics Covered:
✅ Derivation of Cauchy-Riemann equations
✅ Use of the equations to identify non-differentiable points
✅ Examples showing that Cauchy-Riemann equations can hold even when the function is not differentiable
✅ Clarifying the necessary vs. sufficient condition for differentiability

🎯 Best For:
– B.Sc. & M.Sc. Mathematics students
– Engineering Mathematics (B.Tech.)
– CSIR-NET, GATE, IIT-JAM aspirants
– Learners of Complex Analysis and Function Theory

📌 Previous Lecture: [Differentiability of Complex Functions – Watch Here]
📌 Next Lecture: Analytic Functions and Harmonic Conjugates

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