Overview
This lecture covers the Cauchy-Riemann equations, conditions for analyticity and differentiability of complex functions, and verification of these concepts with examples.
Cauchy-Riemann Equations
- For a function ( f(z) = u(x, y) + i v(x, y) ), analyticity requires partial derivatives to exist and satisfy:
- ( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} )
- ( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} )
- The derivative ( f'(z) = \lim_{\Delta z \to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z} ).
- Analyticity requires both continuity and differentiability within a neighborhood around ( z ).
Verification and Examples
- To check CR equations, compute all relevant partial derivatives of ( u ) and ( v ).
- Example function: ( f(z) = x^2 + 2ixy - iy^2 ).
- Real part ( u(x, y) = x^2 - y^2 ), imaginary part ( v(x, y) = 2xy ).
- Compute derivatives: ( u_x = 2x ), ( v_y = 2x ), ( u_y = -2y ), ( v_x = 2y ).
- CR equations are satisfied; thus, ( f(z) ) is analytic.
- Exponential example: ( f(z) = e^x (\cos y + i\sin y) ).
- ( u(x, y) = e^x \cos y ), ( v(x, y) = e^x \sin y ).
- Show ( u_x = v_y ) and ( u_y = -v_x ), confirming analytic nature.
Analyticity and Differentiability
- A function is analytic at ( z ) if CR equations are satisfied and partial derivatives are continuous.
- If a function is analytic in a domain, it is differentiable and its derivative can be found using the formulas from CR equations.
Key Terms & Definitions
- Analytic Function — A complex function that is differentiable in a neighborhood of every point in its domain.
- Cauchy-Riemann Equations — System of equations (\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}) and (\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}) essential for analyticity.
- Partial Derivative — The derivative of a function with respect to one variable, holding others constant.
Action Items / Next Steps
- Practice verifying the Cauchy-Riemann equations for various complex functions.
- Complete any assigned problems on analyticity and differentiability of functions.
- Review the relationship between Cauchy-Riemann equations and differentiability for exam preparation.