Question

Two functions u and v of x and y are said to satisfy the Cauchy - Riemann equations if u = = Vy Uy = -Vr. Show that u(x, y) = x² - y² and v(x, y): and = 2xy safisfy these equations.

Answer 1

The functions u(x, y) = x² - y² and v(x, y) = 2xy satisfy the Cauchy-Riemann equations, as the partial derivatives ux matches vy and uy matches -vx, proving they are the real and imaginary components of an analytic complex function.

Step-by-step explanation:

The question asks us to prove that the functions u(x, y) = x² - y² and v(x, y) = 2xy satisfy the Cauchy-Riemann equations. These equations are a set of two partial differential equations which, if satisfied by two functions, show that they are the real and imaginary parts of an analytic complex function of the form f(z) = u(x, y) + iv(x, y), where z = x + iy.

To show the functions satisfy the Cauchy-Riemann equations, we must compare the partial derivatives of u with respect to x (denoted as ux) and y (denoted as uy), as well as the partial derivatives of v (denoted as vx and vy).

The Cauchy-Riemann equations state that for the functions to be analytic:

• ux = vy
• uy = -vx

Let's calculate the partial derivatives:

• ux = 2x
• uy = -2y
• vx = 2y
• vy = 2x

As we can see:

• ux = 2x = vy
• uy = -2y = -vx

Therefore, the functions satisfy the Cauchy-Riemann equations and can be considered the real and imaginary parts of a complex analytic function.

Answer 2

The functions u(x, y) = x² - y² and v(x, y) = 2xy satisfy the Cauchy-Riemann equations as their partial derivatives u_x = 2x and v_y = 2x are equal, and u_y = -2y and v_x = 2y are negatives of each other.

Step-by-step explanation:

The student has asked about verifying whether two functions satisfy the Cauchy-Riemann equations. The Cauchy-Riemann equations are conditions that must be satisfied for a function of a complex variable to be differentiable. These conditions state that for two functions u(x, y) and v(x, y), the partial derivatives must satisfy ux = vy and uy = -vx. To verify this for the functions given, u(x, y) = x2 - y2 and v(x, y) = 2xy, we calculate the partial derivatives:

• ux = 2x and vy = 2x: they are equal, so the first Cauchy-Riemann condition is satisfied.
• uy = -2y and vx = 2y: they are negatives of each other, so the second Cauchy-Riemann condition is satisfied.

Hence, it is shown that the functions u(x, y) = x2 - y2 and v(x, y) = 2xy satisfy the Cauchy-Riemann equations.