Cauchy-Riemann Equations

The Cauchy–Riemann equations are fundamental in complex analysis and are central to understanding the behavior of analytic functions. These equations provide the necessary conditions for a function to be complex differentiable and, by extension, to be analytic.

Introduction to complex numbers

To understand the Cauchy-Riemann equations, we first need to become familiar with complex numbers. A complex number z is expressed as:

z = x + yi

where x and y are real numbers, and i is an imaginary unit having the property:

i 2 = -1

The real part of the complex number is x (represented by Re(z) = x) and the imaginary part is y (represented by Im(z) = y).

Functions of a complex variable

A function f of a complex variable z is written as:

f(z) = u(x, y) + v(x, y)i

where u and v are real-valued functions of two real variables.

Complex differentiability

For a function f to be differentiable at the point z_0 in the complex plane, the limit:

lim_{{h to 0}} frac{f(z_0 + h) - f(z_0)}{h}

must exist, where h is a complex number near zero.

Cauchy–Riemann equations

The Cauchy-Riemann equations are the conditions that the partial derivatives of the real and imaginary parts of f(z) = u(x, y) + v(x, y)i must satisfy in order for the function f to be differentiable at a point. These equations are:

frac{partial u}{partial x} = frac{partial v}{partial y} frac{partial u}{partial y} = -frac{partial v}{partial x}

If these equations are valid and the partial derivatives are continuous, then f(z) is analytic.

Derivation of the Cauchy–Riemann equations

Let us derive the Cauchy-Riemann equations by considering a complex function f(z) = u(x, y) + v(x, y)i.

The difference of f can be obtained from two different paths: one along x axis and the other along y axis.

When the growth is along x axis, we have:

f(z + Delta x) - f(z) = u(x + Delta x, y) - u(x, y) + i(v(x + Delta x, y) - v(x, y)) approx frac{partial u}{partial x}Delta x + ifrac{partial v}{partial x}Delta x

so,

frac{f(z + Delta x) - f(z)}{Delta x} approx frac{partial u}{partial x} + ifrac{partial v}{partial x}

When the growth is along y axis, the approach is similar:

f(z + iDelta y) - f(z) = u(x, y + Delta y) - u(x, y) + i(v(x, y + Delta y) - v(x, y)) approx frac{partial u}{partial y}Delta y + ifrac{partial v}{partial y}Delta y

Which gives us:

frac{f(z + iDelta y) - f(z)}{iDelta y} approx ileft(frac{partial u}{partial y} + ifrac{partial v}{partial y}right)

so:

frac{f(z + iDelta y) - f(z)}{Delta y} approx -frac{partial v}{partial y} + ifrac{partial u}{partial y}

Since the derivative of f(z) must be path-independent, equating the two expressions gives us the Cauchy-Riemann equations.

Viewing complex variation

Consider a complex function, and imagine a small circle in the domain. If f(z) is differentiable, then the image of this circle through f will be a small ellipse. The orientation of this ellipse is determined by the Cauchy-Riemann conditions.

Examples of analytic functions

Example 1: Polynomial

Consider a polynomial function:

f(z) = z^n = (x + yi)^n

Using the binomial theorem, we can expand this, and see that the Cauchy–Riemann equations apply.

Example 2: Exponential function

Consider the exponential function f(z) = e^z, where e^z = e^x cdot e^{yi}.

Applying Euler's formula, e^{yi} = cos(y) + isin(y), we get:

f(z) = e^x(cos(y) + isin(y))

Here, the real part u(x, y) = e^x cos(y) and the imaginary part v(x, y) = e^x sin(y). By computing the partial derivatives, and applying the Cauchy-Riemann equations, we confirm that the function is analytic everywhere.

Importance of the Cauchy–Riemann equations

The Cauchy-Riemann equations are the gateway into the rich theory of analytic functions in the complex plane. Solutions of these equations reveal amazing insights into the properties of holomorphic functions, such as preservation of angles (conformal mapping) and the existence of Taylor series representations.

Application

Some notable applications of the Cauchy–Riemann equations include their use in physics and engineering, particularly in problems involving fluid dynamics and electrostatics, where potential functions are often analytic.

Conclusion

In summary, the Cauchy-Riemann equations are important in complex analysis, providing necessary and sufficient conditions for analytic functions. Understanding them provides fundamental insight into a wide range of applications in mathematics and science.

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