3.2 Differentiability and analyticity

Complex functions can be tricky, but differentiability and analyticity are key concepts to grasp. They're all about how smoothly a function behaves and whether it can be approximated by polynomials.

Differentiability means a function has a well-defined rate of change at every point. Analyticity is even stronger - it requires the function to be infinitely differentiable and have a power series expansion. These properties are crucial for many advanced techniques in complex analysis.

Derivative of Complex Functions

Definition and Geometric Interpretation

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• The derivative of a complex function $f(z)$ at a point $z_0$ is defined as the limit of the difference quotient $\frac{f(z) - f(z_0)}{z - z_0}$ as $z$ approaches $z_0$, provided the limit exists
• Measures the rate of change of the function at a given point
• Geometrically, represents the local stretching and rotation of the function near that point
  • The argument (angle) of the derivative indicates the angle of rotation
  • The modulus (magnitude) of the derivative represents the scaling factor
• If the derivative exists at a point, the function is said to be differentiable at that point

Properties of Differentiable Functions

• The sum, difference, product, and quotient of differentiable functions are also differentiable, provided the denominator of the quotient is non-zero
• The composition of two differentiable functions is also differentiable
  • The derivative can be found using the chain rule
• Examples of differentiable functions:
  • Polynomial functions ($z^2 + 3z + 1$)
  • Exponential functions ($e^z$)
  • Trigonometric functions ($\sin(z)$, $\cos(z)$)

Differentiability of Complex Functions

Conditions for Differentiability

• A complex function $f(z)$ is differentiable at a point $z_0$ if the limit of the difference quotient $\frac{f(z) - f(z_0)}{z - z_0}$ exists as $z$ approaches $z_0$ from any direction
• For differentiability, the limit of the difference quotient must be the same regardless of the path along which $z$ approaches $z_0$
• If a complex function is differentiable at every point within an open domain, the function is said to be holomorphic or analytic on that domain

Examples of Differentiability

• The function $f(z) = z^2$ is differentiable at every point in the complex plane
  • The derivative is $f'(z) = 2z$
• The function $f(z) = |z|$ is not differentiable at $z = 0$
  • The limit of the difference quotient depends on the path along which $z$ approaches $0$
• The function $f(z) = \overline{z}$ (complex conjugate) is not differentiable at any point
  • The limit of the difference quotient does not exist

Analyticity and Differentiability

Definition and Properties of Analytic Functions

• A complex function is analytic (or holomorphic) on an open domain if it is differentiable at every point within that domain
• Analyticity is a stronger condition than differentiability
  • A function may be differentiable at a point without being analytic in a neighborhood of that point
• Analytic functions possess many desirable properties:
  • Infinitely differentiable
  • Satisfy the Cauchy-Riemann equations
  • Admit contour integral representations
  • Can be represented by a convergent power series in a neighborhood of any point within the domain

Examples of Analytic and Non-Analytic Functions

• The function $f(z) = z^2$ is analytic on the entire complex plane
• The function $f(z) = \overline{z}$ (complex conjugate) is not analytic on any open domain
  • It is differentiable only at $z = 0$
• The function $f(z) = |z|$ (absolute value) is not analytic on any open domain
  • It is not differentiable at $z = 0$

Cauchy-Riemann Equations for Analyticity

Statement and Application of Cauchy-Riemann Equations

• The Cauchy-Riemann equations provide a necessary and sufficient condition for a complex function to be analytic on an open domain
• For a complex function $f(z) = u(x, y) + iv(x, y)$, where $z = x + iy$, the Cauchy-Riemann equations are:
  • $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$
  • $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• To verify analyticity using the Cauchy-Riemann equations:
  1. Express the function in terms of its real and imaginary parts, $u(x, y)$ and $v(x, y)$
  2. Calculate the partial derivatives $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$, $\frac{\partial v}{\partial x}$, and $\frac{\partial v}{\partial y}$
  3. Check if they satisfy the Cauchy-Riemann equations at every point in the domain
  4. Ensure that the partial derivatives are continuous on the domain

Examples of Verifying Analyticity

• The function $f(z) = z^2 = (x + iy)^2 = x^2 - y^2 + i(2xy)$
  • $u(x, y) = x^2 - y^2$ and $v(x, y) = 2xy$
  • $\frac{\partial u}{\partial x} = 2x$, $\frac{\partial u}{\partial y} = -2y$, $\frac{\partial v}{\partial x} = 2y$, $\frac{\partial v}{\partial y} = 2x$
  • The Cauchy-Riemann equations are satisfied, and the partial derivatives are continuous, so $f(z)$ is analytic on the entire complex plane
• The function $f(z) = \overline{z} = x - iy$
  • $u(x, y) = x$ and $v(x, y) = -y$
  • $\frac{\partial u}{\partial x} = 1$, $\frac{\partial u}{\partial y} = 0$, $\frac{\partial v}{\partial x} = 0$, $\frac{\partial v}{\partial y} = -1$
  • The Cauchy-Riemann equations are not satisfied, so $f(z)$ is not analytic on any open domain