Cauchy-Riemann Equations Worked Example
This animation walks through a complex analysis exam problem, finding where f(z) = u(x,y) + iv(x,y) is differentiable and analytic. It separates the real and imaginary parts, computes all four partial derivatives step by step, and applies the Cauchy-Riemann equations to build and solve a system of equations. The visual distinction between differentiability at isolated points versus analyticity in a region is emphasized, making it useful for students preparing for complex variables exams.
16:9 · every frame verified for overlaps, spacing and edges before rendering
The prompt that made it
Create a detailed educational video in Spanish explaining **Exercise 1(a) of this complex analysis exam from beginning to end**, as if you were teaching a student who needs to learn how to solve this exact type of problem for an exam. The exercise is: Sea f(z) = (3x²y² − 1/2 x⁴ − 1/2 y⁴) + i(3y²x − 2x³y) Halle el dominio de derivabilidad de f(z) y represéntelo gráficamente. ¿En qué puntos es f(z) analítica? Follow the solution methodology used in the provided exam exactly. TEACHING GOAL: Do not simply show the final answer. Teach me the general procedure so that I could solve a similar exercise by myself. STRUCTURE THE VIDEO LIKE THIS: 1. INTRODUCTION Explain what the problem is asking: * We have a complex function f(z) = u(x,y) + i v(x,y). * To determine where f is differentiable, we use the Cauchy–Riemann equations. * Explain the difference between: • differentiable at a point • analytic in a neighborhood of a point. 2. IDENTIFY u(x,y) AND v(x,y) Clearly separate the real and imaginary parts: u(x,y) = 3x²y² − 1/2 x⁴ − 1/2 y⁴ v(x,y) = 3x²y − 2x³y Show visually where each expression comes from. 3. CALCULATE ALL FOUR PARTIAL DERIVATIVES Derive, step by step, rather than just displaying the result: u_x u_y v_x v_y Show every differentiation step and simplify carefully. 4. APPLY THE CAUCHY–RIEMANN EQUATIONS Write explicitly: u_x = v_y u_y = −v_x Substitute the derivatives and simplify both equations. Show how the system becomes: x y (y − 1) = 0 and 2x²y + y² − 2x² − 1 = 0 Do not jump directly to these factored equations. Show the algebra that leads to them. 5. SOLVE THE SYSTEM SYSTEMATICALLY Explain that from x y (y − 1) = 0 we have three possibilities: x = 0 y = 0 y = 1 Analyze each case separately. CASE 1: x = 0 Substitute into the second equation. Show that: y² − 1 = 0 therefore: y = ±1 so the points are: z = i z = −i CASE 2: y = 0 Substitute into the second equation. Show that: −2x² − 1 = 0 and explain clearly why this has no real solutions. CASE 3: y = 1 Substitute into the second equation and show explicitly that it becomes an identity. Therefore every point on: y = 1 satisfies the Cauchy–Riemann equations. 6. BUILD THE FINAL DOMAIN OF DIFFERENTIABILITY Combine the solutions and explain visually that: D_der(f) = {−i} ∪ {z ∈ C : Im(z) = 1} Show this on an Argand diagram / Cartesian complex plane: * mark the point −i * draw the horizontal line y = 1 * clearly label both. 7. EXPLAIN ANALYTICITY CAREFULLY This is very important. Explain that satisfying the Cauchy–Riemann equations at isolated points is enough for differentiability at those points here because all partial derivatives are continuous, but it does NOT automatically make the function analytic there. Then explain the neighborhood requirement: A function is analytic at a point if it is differentiable in some neighborhood of that point. For any point on y = 1, an arbitrarily small neighborhood contains points not belonging to the differentiability set. The same is true for z = −i. Therefore the function is not analytic at any point. Conclude: D_an(f) = ∅ 8. FINAL EXAM SUMMARY End with a compact "how to solve this type of exercise" checklist: 9. Separate f into u and v. 10. Compute u_x, u_y, v_x, v_y. 11. Apply Cauchy–Riemann. 12. Solve the resulting system. 13. Represent the differentiability set graphically. 14. For analyticity, check whether differentiability holds in an entire neighborhood. IMPORTANT STYLE: * Spanish. * University-level but very clear. * Assume I understand basic algebra but may be confused about complex differentiability. * Prioritize intuition and visual explanation. * Show every important algebraic step. * Use clean mathematical typography. * Animate the equations as they are derived. * Use an Argand/Cartesian plane for the geometric interpretation. * Do not omit the distinction between differentiability and analyticity. * Do not introduce methods that are not necessary for this exercise. * Do not merely read the answer; actually teach the reasoning. * Finish by showing the complete solution on screen one final time.
Make the next one in this series
- Show how to verify analyticity using polar form Cauchy-Riemann equations
- Explain how to find harmonic conjugates once analyticity is established
- Demonstrate a similar exercise where the domain of differentiability is an entire curve
