SUMMARY
The function f(z) = Re(z)Im(z) + i Im(z) is differentiable only at the point z = 0 in the complex plane. By applying the Cauchy-Riemann equations, it is established that the necessary conditions for differentiability are satisfied solely at this point. The real part of the function can be expressed as u = xy and the imaginary part as v = y, leading to the conclusion that differentiability fails elsewhere.
PREREQUISITES
- Understanding of complex functions and their properties
- Familiarity with Cauchy-Riemann equations
- Knowledge of real and imaginary parts of complex numbers
- Basic calculus concepts related to differentiability
NEXT STEPS
- Study the Cauchy-Riemann equations in detail
- Explore differentiability of complex functions
- Investigate examples of functions that are differentiable at multiple points
- Learn about the implications of differentiability in complex analysis
USEFUL FOR
Students studying complex analysis, mathematicians interested in differentiability, and educators teaching the properties of complex functions.