Math Problem Statement

state and prove cr equations in cartesian form

Solution

Cartesian Form of Cauchy-Riemann Equations

The Cauchy-Riemann equations are a set of two equations that provide a condition for a complex function to be differentiable (or holomorphic) in the complex plane. They are fundamental in complex analysis and establish the relationship between the real and imaginary parts of a complex function.

Let f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y) be a complex function, where u(x,y)u(x, y) and v(x,y)v(x, y) are the real and imaginary parts of the complex function, respectively, and z=x+iyz = x + iy with xx and yy being real variables.

The Cauchy-Riemann equations in Cartesian form are:

ux=vy\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} uy=vx\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

These equations express the fact that the real part u(x,y)u(x, y) and the imaginary part v(x,y)v(x, y) of a complex function must satisfy these two relationships for f(z)f(z) to be differentiable.

Proof of the Cauchy-Riemann Equations

Step 1: Differentiability of f(z)f(z)

Let the complex function f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y) be given, where z=x+iyz = x + iy and x,yx, y are real variables.

The function f(z)f(z) is differentiable at a point if the limit:

limΔz0f(z+Δz)f(z)Δz\lim_{\Delta z \to 0} \frac{f(z + \Delta z) - f(z)}{\Delta z}

exists, where Δz\Delta z is a small change in the complex variable zz, which can be written as:

Δz=Δx+iΔy\Delta z = \Delta x + i \Delta y

Expanding f(z)f(z), we have:

f(z+Δz)=u(x+Δx,y+Δy)+iv(x+Δx,y+Δy)f(z + \Delta z) = u(x + \Delta x, y + \Delta y) + i v(x + \Delta x, y + \Delta y)

Now, using the definition of the derivative, we compute the change in f(z)f(z):

[ \frac{f(z + \Delta z) - f(z)}{\Delta z} = \frac{\left[ u(x + \Delta x, y + \Delta y) - u(x, y) \right] + i \left[ v(x + \Delta x, y + \Delta y) - v(x, y) \right]}{\Delta x + i \Delta y} ]

Step 2: Expanding the Real and Imaginary Parts

The real and imaginary parts of the difference are computed as:

Δu=u(x+Δx,y+Δy)u(x,y)\Delta u = u(x + \Delta x, y + \Delta y) - u(x, y) Δv=v(x+Δx,y+Δy)v(x,y)\Delta v = v(x + \Delta x, y + \Delta y) - v(x, y)

Thus, we need to examine the limits of the real and imaginary parts of the complex difference as Δz0\Delta z \to 0. For the limit to exist (which means f(z)f(z) is differentiable), the behavior of the real and imaginary parts must satisfy certain conditions.

Step 3: Limit Analysis and Conclusion

For the limit to exist in the complex plane, the change in the real part uu must be related to the change in the imaginary part vv in a way that the directional derivatives in the real and imaginary directions are consistent.

By calculating the limit as Δz0\Delta z \to 0, we derive the following relationships:

ux=vyanduy=vx\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

These are precisely the Cauchy-Riemann equations in Cartesian form, which must hold for f(z)f(z) to be differentiable.

Conclusion

The Cauchy-Riemann equations in Cartesian form are:

ux=vy\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} uy=vx\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

These equations are a necessary condition for the differentiability of a complex function. If they are satisfied, the function f(z)f(z) is holomorphic (complex differentiable) at that point.

Would you like further details on the Cauchy-Riemann equations or their application in complex analysis?

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Math Problem Analysis

Mathematical Concepts

Complex Analysis
Differentiability
Cauchy-Riemann Equations

Formulas

∂u/∂x = ∂v/∂y
∂u/∂y = -∂v/∂x

Theorems

Cauchy-Riemann Equations

Suitable Grade Level

University/Graduate Level