2.2 Mappings by elementary functions

Complex functions map points from one complex plane to another. Elementary functions like linear, exponential, and trigonometric play a crucial role. They exhibit unique properties and behaviors, forming the foundation for more advanced concepts in complex analysis.

Understanding these mappings is essential for visualizing and manipulating complex functions. From simple linear transformations to intricate conformal mappings, these tools allow us to explore the rich landscape of complex analysis and its applications in various fields.

Mappings of Elementary Functions

Linear and Special Functions

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• Linear functions, such as $f(z) = az + b$ where $a$ and $b$ are complex constants, map lines to lines and circles to circles in the complex plane
  • The value of $a$ determines the rotation and scaling, while $b$ represents the translation
• The identity function, $f(z) = z$, maps the complex plane onto itself without any transformation
• The conjugate function, $f(z) = \bar{z}$, reflects the complex plane across the real axis, mapping points $(x, y)$ to $(x, -y)$

Visualizing Complex Mappings

• Elementary functions in complex analysis include exponential, logarithmic, trigonometric, polynomial, and rational functions
  • These functions map points from the complex plane to another set of points in the complex plane
• Visualizing complex functions requires considering both the domain (input) and codomain (output) as two-dimensional planes
  • The input is the complex $z$-plane and the output is the complex $w$-plane

Properties of Complex Functions

Exponential and Logarithmic Functions

• The complex exponential function, $e^z = e^{x+iy} = e^x * (\cos(y) + i*\sin(y))$, is periodic along the imaginary axis with period $2\pi i$ and unbounded along the real axis
• The complex logarithm, $\log(z)$, is a multi-valued function with infinitely many branches
  • The principal branch, denoted as $\Log(z)$, is defined by restricting the imaginary part to the interval $(-\pi, \pi]$

Trigonometric and Hyperbolic Functions

• Complex trigonometric functions, such as $\sin(z)$, $\cos(z)$, and $\tan(z)$, can be defined using the complex exponential function through Euler's formula: $e^{iz} = \cos(z) + i*\sin(z)$
  • These functions exhibit periodicity in both the real and imaginary parts, with the period being related to $\pi$
  • The complex sine and cosine functions have zeros at integer multiples of $\pi$ and $\pi/2$, respectively, along the real axis
• The hyperbolic trigonometric functions, $\sinh(z)$, $\cosh(z)$, and $\tanh(z)$, can be defined using the exponential function and have similar properties to their real counterparts

Mappings of Polynomial and Rational Functions

Polynomial Functions

• Polynomial functions, $P(z) = a_n*z^n + a_{n-1}*z^{n-1} + ... + a_1*z + a_0$, where the coefficients $a_i$ are complex numbers, map the complex plane to itself
  • The degree of the polynomial determines the number of zeros (roots) and the behavior at infinity

Rational Functions and Critical Points

• Rational functions, $R(z) = P(z) / Q(z)$, where $P(z)$ and $Q(z)$ are polynomial functions, map the complex plane to itself, except at the poles (zeros of the denominator)
  • These functions can have zeros, poles, and branch points
• Critical points of a complex function are points where the derivative is zero or does not exist
  • These points can be classified as zeros ($f(z) = 0$), poles ($f(z)$ approaches infinity), or branch points (multi-valued points)
  • The order of a zero or pole determines the behavior of the function near the critical point
    • A zero of order $m$ means $f(z) \sim (z - z_0)^m$ near the point $z_0$
    • A pole of order $n$ means $f(z) \sim 1 / (z - z_0)^n$
• The residue of a complex function at a pole is the coefficient of the $(z - z_0)^{-1}$ term in the Laurent series expansion around the pole
  • Residues are useful in evaluating complex integrals using the residue theorem

Conformal Mappings for Geometric Properties

Conformal Mappings and Analytic Functions

• Conformal mappings are angle-preserving transformations that map a domain in the complex plane to another domain while preserving local angles and shapes of infinitesimal figures
• Analytic functions (functions that are differentiable at every point in a domain) are conformal at points where the derivative is non-zero
  • The Cauchy-Riemann equations provide a necessary and sufficient condition for a complex function to be analytic (and thus conformal) in a domain
• Examples of conformal mappings include linear functions, exponential functions, and the complex logarithm (in a restricted domain)

Applications and the Riemann Mapping Theorem

• Conformal mappings can be used to solve problems in physics and engineering, such as in fluid dynamics, electrostatics, and heat transfer
  • They transform complicated geometries into simpler ones while preserving the underlying physical properties
• The Riemann mapping theorem states that any simply connected domain in the complex plane (other than the entire plane) can be conformally mapped onto the open unit disk
  • This theorem has significant implications in the study of complex analysis and its applications