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 be a complex function, where and are the real and imaginary parts of the complex function, respectively, and with and being real variables.
The Cauchy-Riemann equations in Cartesian form are:
These equations express the fact that the real part and the imaginary part of a complex function must satisfy these two relationships for to be differentiable.
Proof of the Cauchy-Riemann Equations
Step 1: Differentiability of
Let the complex function be given, where and are real variables.
The function is differentiable at a point if the limit:
exists, where is a small change in the complex variable , which can be written as:
Expanding , we have:
Now, using the definition of the derivative, we compute the change in :
[ \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:
Thus, we need to examine the limits of the real and imaginary parts of the complex difference as . For the limit to exist (which means 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 must be related to the change in the imaginary part in a way that the directional derivatives in the real and imaginary directions are consistent.
By calculating the limit as , we derive the following relationships:
These are precisely the Cauchy-Riemann equations in Cartesian form, which must hold for to be differentiable.
Conclusion
The Cauchy-Riemann equations in Cartesian form are:
These equations are a necessary condition for the differentiability of a complex function. If they are satisfied, the function is holomorphic (complex differentiable) at that point.
Would you like further details on the Cauchy-Riemann equations or their application in complex analysis?
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Understanding the Cauchy-Riemann Equations in Complex Analysis
Simplifying the Cauchy-Riemann Equations for Holomorphic Functions
Cauchy-Riemann Equations in Polar Coordinates: Conditions for Holomorphic Functions
Determining Functions that Satisfy the Cauchy-Riemann Equations in Complex Analysis
Verify Cauchy-Riemann Equations for f(z) = e^x (cos y - i sin y)