2.3 Cauchy-Riemann equations

The Cauchy-Riemann equations are crucial for understanding complex differentiability. They provide a necessary condition for a function to be holomorphic, linking the real and imaginary parts of complex functions through partial derivatives.

These equations have significant implications in complex analysis. They're key to conformal mappings, harmonic functions, and analytic functions. Understanding them is essential for grasping more advanced concepts in complex integration and contour integrals.

Definition of Cauchy-Riemann equations

• The Cauchy-Riemann equations are a set of partial differential equations that establish a necessary condition for a complex-valued function to be complex differentiable (holomorphic)
• For a complex function $f(z) = u(x, y) + iv(x, y)$, where $u$ and $v$ are real-valued functions, the Cauchy-Riemann equations are: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• Satisfying the Cauchy-Riemann equations is a necessary but not sufficient condition for a function to be complex differentiable

Cauchy-Riemann equations for differentiability

Real and imaginary parts

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• The Cauchy-Riemann equations relate the partial derivatives of the real and imaginary parts of a complex function
• For a complex function $f(z) = u(x, y) + iv(x, y)$, $u(x, y)$ represents the real part and $v(x, y)$ represents the imaginary part
• The equations ensure that the real and imaginary parts are interconnected in a way that allows for complex differentiability

Partial derivatives

• The Cauchy-Riemann equations involve partial derivatives of the real and imaginary parts with respect to $x$ and $y$
• $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ denote the partial derivatives of the real part $u$ with respect to $x$ and $y$, respectively
• $\frac{\partial v}{\partial x}$ and $\frac{\partial v}{\partial y}$ denote the partial derivatives of the imaginary part $v$ with respect to $x$ and $y$, respectively
• These partial derivatives must satisfy the Cauchy-Riemann equations for the complex function to be differentiable

Geometric interpretation

Conformal mappings

• Functions that satisfy the Cauchy-Riemann equations are called conformal mappings
• Conformal mappings preserve angles between curves in the complex plane
• They locally preserve the shape of infinitesimal figures, although the size may change

Angle preservation

• One of the key properties of conformal mappings is angle preservation
• If two curves intersect at a certain angle in the domain of a conformal mapping, their images under the mapping will intersect at the same angle
• This property is a consequence of the Cauchy-Riemann equations

Local rotations and dilations

• Conformal mappings can be interpreted as a combination of local rotations and dilations (scaling) in the complex plane
• The Cauchy-Riemann equations ensure that the mapping locally behaves like a rotation and uniform scaling
• The angle of rotation and the scale factor may vary from point to point

Polar form of Cauchy-Riemann equations

• The Cauchy-Riemann equations can be expressed in polar coordinates $(r, \theta)$
• For a complex function $f(z) = u(r, \theta) + iv(r, \theta)$ in polar form, the Cauchy-Riemann equations become: $\frac{\partial u}{\partial r} = \frac{1}{r}\frac{\partial v}{\partial \theta}$ and $\frac{1}{r}\frac{\partial u}{\partial \theta} = -\frac{\partial v}{\partial r}$
• The polar form is useful when working with functions expressed in terms of modulus and argument

Harmonic functions and Cauchy-Riemann equations

Laplace's equation

• A function $u(x, y)$ is called harmonic if it satisfies Laplace's equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
• If a complex function $f(z) = u(x, y) + iv(x, y)$ satisfies the Cauchy-Riemann equations, then both $u(x, y)$ and $v(x, y)$ are harmonic functions

Conjugate harmonic functions

• If $u(x, y)$ and $v(x, y)$ form a pair of functions satisfying the Cauchy-Riemann equations, they are called conjugate harmonic functions
• Given a harmonic function $u(x, y)$, its harmonic conjugate $v(x, y)$ can be found using the Cauchy-Riemann equations
• The complex function $f(z) = u(x, y) + iv(x, y)$ formed by a pair of conjugate harmonic functions is holomorphic (complex differentiable)

Cauchy-Riemann equations and analyticity

Holomorphic functions

• A complex function $f(z)$ is called holomorphic (or analytic) in a domain $D$ if it is complex differentiable at every point in $D$
• Holomorphic functions satisfy the Cauchy-Riemann equations in their domain
• Examples of holomorphic functions include polynomials, exponential functions, and trigonometric functions

Complex differentiability vs real differentiability

• Complex differentiability is a stronger condition than real differentiability
• A complex function that is complex differentiable (holomorphic) is always real differentiable, but the converse is not true
• Real differentiability does not imply the Cauchy-Riemann equations are satisfied, while complex differentiability does

Applications of Cauchy-Riemann equations

Complex integration

• The Cauchy-Riemann equations play a crucial role in complex integration theory
• They are used to derive important results such as the Cauchy integral formula and the residue theorem
• The Cauchy integral formula relates the value of a holomorphic function inside a closed contour to the values of the function on the contour

Contour integrals

• Contour integrals are integrals of complex functions along closed paths in the complex plane
• The Cauchy-Riemann equations allow for the evaluation of certain contour integrals using techniques such as the Cauchy integral formula and the residue theorem
• These techniques simplify the calculation of integrals that would be difficult or impossible to evaluate using real integration methods

Residue theorem

• The residue theorem is a powerful tool for evaluating contour integrals of complex functions with singularities (poles)
• It states that the contour integral of a meromorphic function (holomorphic except for poles) around a closed path is equal to $2\pi i$ times the sum of the residues enclosed by the path
• The Cauchy-Riemann equations are essential in the derivation and application of the residue theorem

Examples and exercises

Verifying Cauchy-Riemann equations

• Example: Check if the function $f(z) = z^2$ satisfies the Cauchy-Riemann equations
  • Solution: $f(z) = (x + iy)^2 = x^2 - y^2 + i(2xy)$, so $u(x, y) = x^2 - y^2$ and $v(x, y) = 2xy$
  • $\frac{\partial u}{\partial x} = 2x$, $\frac{\partial v}{\partial y} = 2x$, $\frac{\partial u}{\partial y} = -2y$, $\frac{\partial v}{\partial x} = 2y$
  • The Cauchy-Riemann equations are satisfied, so $f(z)$ is holomorphic

Finding harmonic conjugates

• Example: Find the harmonic conjugate of $u(x, y) = e^x \cos y$
  • Solution: Using the Cauchy-Riemann equations, $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
  • $\frac{\partial u}{\partial x} = e^x \cos y$, so $\frac{\partial v}{\partial y} = e^x \cos y$, which implies $v(x, y) = e^x \sin y + C(x)$
  • $\frac{\partial u}{\partial y} = -e^x \sin y$, so $-\frac{\partial v}{\partial x} = -e^x \sin y$, which implies $C(x) = 0$
  • The harmonic conjugate is $v(x, y) = e^x \sin y$

Proving holomorphicity

• Example: Prove that the function $f(z) = \sin z$ is holomorphic
  • Solution: $f(z) = \sin(x + iy) = \sin x \cosh y + i \cos x \sinh y$, so $u(x, y) = \sin x \cosh y$ and $v(x, y) = \cos x \sinh y$
  • $\frac{\partial u}{\partial x} = \cos x \cosh y$, $\frac{\partial v}{\partial y} = \cos x \cosh y$, $\frac{\partial u}{\partial y} = \sin x \sinh y$, $\frac{\partial v}{\partial x} = -\sin x \sinh y$
  • The Cauchy-Riemann equations are satisfied, so $f(z)$ is holomorphic