Analytic Function and the Cauchy-Riemann Equations: Exploring the Complex Function uxy x^3 - 3xy^2y

Introduction

In the field of complex analysis, an analytic function is a function that is locally given by a convergent power series. This function plays a vital role in various branches of mathematics, including physics and engineering. One way to determine whether a given function is analytic is to check if it satisfies the Cauchy-Riemann equations. In this article, we will explore the analytic function with the real part u(x, y) x^3 - 3xy^2y and demonstrate how to find its imaginary part using the Cauchy-Riemann equations.

The Cauchy-Riemann Equations and Their Application

The Cauchy-Riemann equations are a pair of partial differential equations which must be satisfied if a function is to be complex differentiable. These equations are given by:

1. (#8706;u/#8706;x #8706;v/#8706;y)

2. (#8706;u/#8706;y -#8706;v/#8706;x)

Given the function with the real part u(x, y) x^3 - 3xy^2y, we will find its imaginary part and verify the Cauchy-Riemann equations.

Part 1: Determining the Partial Derivatives

First, let's find the partial derivatives of the real part u(x, y):

(frac{partial u}{partial x} 3x^2 - 3y^2)

(frac{partial u}{partial y} 1 - 6xy)

According to the Cauchy-Riemann equations:

(frac{partial u}{partial x} frac{partial v}{partial y} Rightarrow frac{partial v}{partial y} 3x^2 - 3y^2)

(frac{partial u}{partial y} -frac{partial v}{partial x} Rightarrow -frac{partial v}{partial x} 1 - 6xy)

Part 2: Integrating to Find the Imaginary Part

Now, we will integrate the partial derivatives to find the imaginary part v(x, y):

Integrating (frac{partial v}{partial y} 3x^2 - 3y^2) with respect to y:

(v(x, y) 3x^2y - y^3 f(x))

where (f(x)) is the constant of integration with respect to y.

Integrating (-frac{partial v}{partial x} 6xy - 1) with respect to x:

(f(x) -3x^2 g(y))

where (g(y)) is the constant of integration with respect to x.

Substituting (f(x) -3x^2 g(y)) back into the expression for (v(x, y)), we get:

(v(x, y) 3x^2y - y^3 - 3x^2 g(y))

Since (g(y)) is a constant with respect to x, we can drop it and obtain the imaginary part of the function:

(v(x, y) 3x^2y - x - y^3)

Part 3: Constructing the Analytic Function

Now that we have the real and imaginary parts, we can form the analytic function (f(z) u(x, y) iv(x, y)):

(f(z) x^3 - 3xy^2y i(3x^2y - x - y^3))

Collecting like terms together, we can express the function on the right side in terms of (z x iy):

(f(z) z^3 - iz iC)

Conclusion

In conclusion, we have explored the process of determining the analytic function using the real part (u(x, y) x^3 - 3xy^2y) and the Cauchy-Riemann equations. By finding the imaginary part (v(x, y) 3x^2y - x - y^3 C), we have constructed the complete analytic function:

(f(z) z^3 - iz iC)

This not only satisfies the Cauchy-Riemann equations but also demonstrates the power of complex analysis in understanding the behavior of multi-variable functions.