The are key to understanding complex . They give us a way to check if a function is analytic, which is super important in complex analysis. These equations link the real and imaginary parts of a complex function.
By using these equations, we can figure out if a function is analytic and find harmonic conjugates. This helps us solve all sorts of problems in math and physics, like fluid dynamics and electrostatics. It's a powerful tool for working with complex functions.
Cauchy-Riemann Equations
Cartesian and Polar Forms
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The Cauchy-Riemann equations provide a necessary and sufficient condition for a complex function to be complex differentiable (analytic) at a point
In Cartesian form, for a complex function f(z)=[u(x,y)](https://www.fiveableKeyTerm:u(x,y))+iv(x,y), the Cauchy-Riemann equations are:
∂x∂u=∂y∂v
∂y∂u=−∂x∂v
In polar form, for a complex function f(z)=u(r,θ)+iv(r,θ), the Cauchy-Riemann equations are:
∂r∂u=r1∂θ∂v
∂r∂v=−r1∂θ∂u
The Cauchy-Riemann equations relate the partial derivatives of the real and imaginary parts of a complex function
Derivation from Differentiability
A complex function f(z)=u(x,y)+iv(x,y) is differentiable at a point z0 if the limit of z−z0f(z)−f(z0) exists as z approaches z0, independent of the path along which z approaches z0
Applying the limit definition of the derivative to the real and imaginary parts of f(z) separately leads to the Cauchy-Riemann equations
The derivation involves considering the limit along the real and imaginary axes and equating the corresponding components, resulting in the partial derivative relations
The derivation can also be performed using the polar form of the complex function, leading to the polar form of the Cauchy-Riemann equations
The existence of the complex derivative implies the existence and continuity of the partial derivatives satisfying the Cauchy-Riemann equations
Differentiability and Cauchy-Riemann Equations
Analyticity and Differentiability
A complex function is analytic (complex differentiable) at a point if it is differentiable in a neighborhood of that point
For a complex function to be analytic at a point or in a region, the Cauchy-Riemann equations must be satisfied at that point or throughout the region
If the Cauchy-Riemann equations are satisfied and the partial derivatives are continuous at a point, then the function is analytic at that point
If the Cauchy-Riemann equations are satisfied throughout a region and the partial derivatives are continuous in that region, then the function is analytic in that region
Analyticity is a stronger condition than differentiability, as it requires the function to be differentiable in a neighborhood of a point
Checking Analyticity
To determine if a complex function is analytic, compute the partial derivatives of the real and imaginary parts and check them against the Cauchy-Riemann equations
Example: For f(z)=z2=(x+iy)2=(x2−y2)+i(2xy), we have:
The Cauchy-Riemann equations are satisfied, and the partial derivatives are continuous everywhere, so f(z)=z2 is analytic in the entire complex plane
Example: For f(z)=zˉ=x−iy, we have:
u(x,y)=x, v(x,y)=−y
∂x∂u=1, ∂y∂v=−1
∂y∂u=0, ∂x∂v=0
The Cauchy-Riemann equations are not satisfied, so f(z)=zˉ is not analytic anywhere
Analyticity Using Cauchy-Riemann Equations
Harmonic Functions and Conjugates
A real-valued function u(x,y) is called harmonic if it satisfies Laplace's equation: ∂x2∂2u+∂y2∂2u=0
If u(x,y) is a harmonic function, then there exists a harmonic conjugate v(x,y) such that f(z)=u(x,y)+iv(x,y) is analytic
The harmonic conjugate can be found by integrating the Cauchy-Riemann equations:
v(x,y)=∫∂x∂udy+C(x) or v(x,y)=−∫∂y∂udx+C(y)
Example: If u(x,y)=excosy, then v(x,y)=exsiny is its harmonic conjugate, and f(z)=excosy+iexsiny=ez is analytic
Laplace's Equation
The Cauchy-Riemann equations can be used to derive Laplace's equation in two dimensions
If f(z)=u(x,y)+iv(x,y) is analytic, then both u(x,y) and v(x,y) satisfy Laplace's equation:
∂x2∂2u+∂y2∂2u=0 and ∂x2∂2v+∂y2∂2v=0
This property is useful in solving boundary value problems in various fields, such as fluid dynamics and electrostatics
Example: In electrostatics, the electric potential ϕ(x,y) satisfies Laplace's equation in charge-free regions, and the electric field components can be found using the Cauchy-Riemann equations:
Ex=−∂x∂ϕ and Ey=−∂y∂ϕ
Applications of Cauchy-Riemann Equations
Analytic Function Properties
The Cauchy-Riemann equations can be used to prove properties of analytic functions, such as the Cauchy-Riemann theorem and the Cauchy integral formula
Cauchy-Riemann Theorem: If f(z) is analytic in a simply connected domain D and C is a simple closed curve in D, then ∮Cf(z)dz=0
Cauchy Integral Formula: If f(z) is analytic in a simply connected domain D and C is a simple closed curve in D enclosing a point z0, then f(z0)=2πi1∮Cz−z0f(z)dz
These theorems are fundamental in complex analysis and have numerous applications in mathematics and physics
Boundary Value Problems
The Cauchy-Riemann equations can be used to solve boundary value problems in various fields, such as fluid dynamics, electrostatics, and heat transfer
In these problems, the solution is often a complex function whose real and imaginary parts satisfy certain boundary conditions
Example: In fluid dynamics, the complex potential W(z)=ϕ(x,y)+iψ(x,y) is used to describe the flow of an ideal fluid, where ϕ(x,y) is the velocity potential and ψ(x,y) is the stream function
The Cauchy-Riemann equations relate the velocity components to the potential and stream functions: ux=∂x∂ϕ=∂y∂ψ and uy=∂y∂ϕ=−∂x∂ψ
Boundary conditions on the velocity components or the stream function can be used to determine the complex potential and solve for the flow field
Example: In electrostatics, the complex potential W(z)=ϕ(x,y)+iψ(x,y) is used to describe the electric field in two dimensions, where ϕ(x,y) is the electric potential and ψ(x,y) is the electric flux function
The Cauchy-Riemann equations relate the electric field components to the potential and flux functions: Ex=−∂x∂ϕ=∂y∂ψ and Ey=−∂y∂ϕ=−∂x∂ψ
Boundary conditions on the electric potential or the flux function can be used to determine the complex potential and solve for the electric field