. Suppose f(z) = u(x, y) + iv(x, y) is analytic on a domain D. Show that ∂ 2 ∂x2 |f| 2 + ∂ 2 ∂y2 |f| 2 = 4|f ′ (z)| 2 . on D.
Added by Popagano M.
Step 1
This implies that \( u \) and \( v \) satisfy the Cauchy-Riemann equations: \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}. \] Show more…
Show all steps
Close
Your feedback will help us improve your experience
Kirsty Gledhill and 77 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Let f, g : R -> R be two differentiable functions such that f' = g, f(0) = 0, g' = -f, and g(0) = 1. Show that for all x in R we have (f(x))^2 + (g(x))^2 = 1.
Madhur L.
Let the functions $u(x, y)$ and $v(x, y)$ have continuous second partial derivatives and satisfy the Cauchy-Riemann equations $$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$$ and $$\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$$ Show that $u$ and $v$ are both harmonic.
Partial Differentiation
Higher-Order Derivatives
Verify that u(x, y) = x^2 + 4x - y^2 + 2y is harmonic and find a) the harmonic conjugate function v(x, y) of u(x, y). b) the corresponding analytic function f(z). (f = u + iv)
Adi S.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD