Glossary

2.2 Mappings by elementary functions

4 min read•august 13, 2024

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+bf(z) = az + b where aa and bb are complex constants, map lines to lines and circles to circles in the complex plane
    • The value of aa determines the rotation and scaling, while bb represents the translation
  • The identity function, f(z)=zf(z) = z, maps the complex plane onto itself without any transformation
  • The conjugate function, f(z)=zˉf(z) = \bar{z}, reflects the complex plane across the real axis, mapping points (x,y)(x, y) to (x,−y)(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 zz-plane and the output is the complex ww-plane

Properties of Complex Functions

Exponential and Logarithmic Functions

  • The complex , ez=ex+iy=ex∗(cos⁡(y)+i∗sin⁡(y))e^z = e^{x+iy} = e^x * (\cos(y) + i*\sin(y)), is periodic along the imaginary axis with period 2πi2\pi i and unbounded along the real axis
  • The , log⁡(z)\log(z), is a multi-valued function with infinitely many branches
    • The principal branch, denoted as \Log(z)\Log(z), is defined by restricting the imaginary part to the interval (−π,π](-\pi, \pi]

Trigonometric and Hyperbolic Functions

  • Complex trigonometric functions, such as sin⁡(z)\sin(z), cos⁡(z)\cos(z), and tan⁡(z)\tan(z), can be defined using the complex exponential function through Euler's formula: eiz=cos⁡(z)+i∗sin⁡(z)e^{iz} = \cos(z) + i*\sin(z)
    • These functions exhibit periodicity in both the real and imaginary parts, with the period being related to π\pi
    • The complex sine and cosine functions have zeros at integer multiples of π\pi and π/2\pi/2, respectively, along the real axis
  • The hyperbolic trigonometric functions, sinh⁡(z)\sinh(z), cosh⁡(z)\cosh(z), and tanh⁡(z)\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)=an∗zn+an−1∗zn−1+...+a1∗z+a0P(z) = a_n*z^n + a_{n-1}*z^{n-1} + ... + a_1*z + a_0, where the coefficients aia_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)R(z) = P(z) / Q(z), where P(z)P(z) and Q(z)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)=0f(z) = 0), poles (f(z)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 mm means f(z)∼(z−z0)mf(z) \sim (z - z_0)^m near the point z0z_0
      • A pole of order nn means f(z)∼1/(z−z0)nf(z) \sim 1 / (z - z_0)^n
  • The residue of a complex function at a pole is the coefficient of the (z−z0)−1(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 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 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

Key Terms to Review (18)

Bernhard Riemann: Bernhard Riemann was a German mathematician who made significant contributions to various fields including complex analysis, differential geometry, and mathematical physics. His work laid the groundwork for the development of many important concepts, such as Riemann surfaces and the Riemann mapping theorem, which connect complex functions to geometric structures.
Branch Cuts: Branch cuts are lines or curves in the complex plane that are introduced to define a single-valued branch of a multi-valued function, such as the logarithm or square root. By creating these cuts, we can avoid ambiguity when determining the function's values, particularly as we navigate around points where the function would otherwise loop back on itself. This helps us make sense of mappings created by elementary functions and ensures consistent and well-defined outputs for complex inputs.
Carl Friedrich Gauss: Carl Friedrich Gauss was a renowned German mathematician and physicist known for his contributions to various fields including number theory, statistics, and complex analysis. His work laid the foundation for many concepts in mathematics, particularly regarding the imaginary unit and the properties of complex numbers, which have profound implications in various mathematical mappings and transformations.
Cauchy-Riemann Equations: The Cauchy-Riemann equations are a set of two partial differential equations that provide necessary and sufficient conditions for a complex function to be differentiable at a point in the complex plane. These equations establish a relationship between the real and imaginary parts of a complex function, connecting them to the concept of analyticity and ensuring that the function behaves nicely under differentiation, which is crucial in various areas such as complex exponentials, conformal mappings, and transformations.
Complex logarithm: The complex logarithm is a multi-valued function that extends the concept of the logarithm to complex numbers. It defines the logarithm of a complex number in terms of its magnitude and argument, leading to a result of the form $$ ext{log}(z) = ext{log}|z| + i heta$$ where $$ heta$$ is the argument of the complex number. Understanding this function involves exploring how it behaves under transformations, its properties related to complex exponentials, and its significance in complex analysis.
Conformal Mapping: Conformal mapping is a technique in complex analysis that preserves angles and the local shape of small figures during transformation. This concept connects beautifully with various mathematical structures and functions, allowing for the simplification of complex shapes into more manageable forms, while maintaining critical geometric properties. It plays a crucial role in understanding fluid dynamics, electromagnetic fields, and other physical phenomena where preserving angles is essential.
Exponential Function: An exponential function is a mathematical function of the form $$f(z) = a e^{bz}$$, where $$a$$ and $$b$$ are constants, $$e$$ is Euler's number (approximately 2.71828), and $$z$$ is a complex variable. This function is significant because it models growth and decay processes and has unique properties like continuity and differentiability, connecting deeply with other concepts such as mappings, poles, transforms, and series expansions.
Holomorphic Mapping: A holomorphic mapping is a function that is complex differentiable at every point in its domain. This property of being complex differentiable implies that the function is smooth, continuous, and can be represented by a power series within a certain radius. Holomorphic mappings are foundational in complex analysis, particularly when discussing how elementary functions behave in the complex plane.
Injectivity: Injectivity refers to a property of a function where each element in the codomain is mapped by at most one element from the domain. This means that no two different inputs produce the same output, ensuring that the function maintains distinctness across its mapping. Understanding injectivity is crucial for analyzing functions, especially when determining their behavior and characteristics in complex analysis.
Linear Transformation: A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that if you take any two vectors and combine them through addition or multiply them by a scalar, the result after applying the linear transformation will be the same as applying the transformation to each vector first and then combining them. This property is fundamental in various mathematical contexts, allowing us to analyze how functions behave under transformation and providing insights into their geometric interpretations.
Mapping of the unit disk: The mapping of the unit disk refers to the process of transforming points from the open unit disk in the complex plane, defined as the set of complex numbers $$z$$ such that $$|z| < 1$$, to another domain through various functions. This concept is crucial in understanding how complex functions can reshape regions in the complex plane and provides insight into properties like conformality, which preserves angles, and how singularities behave within the unit disk. It serves as a foundation for exploring more complex mappings by elementary functions.
Mapping onto the Riemann Sphere: Mapping onto the Riemann Sphere refers to the process of extending complex functions to include a point at infinity, allowing for a more comprehensive view of complex analysis. This geometric representation allows for the visualization of complex numbers and their transformations, providing insight into behaviors like singularities and limits. It helps bridge the gap between finite complex numbers and their behavior at infinity, enriching the understanding of functions.
Moebius Transformation: A Moebius transformation is a function that maps the complex plane to itself and is defined by the formula $$f(z) = \frac{az + b}{cz + d}$$ where $a$, $b$, $c$, and $d$ are complex numbers and $ad - bc \neq 0$. This transformation is significant in complex analysis as it represents an important class of functions that preserve angles and the structure of the complex plane. It has wide applications, including in the study of Riemann surfaces and conformal mappings.
Pointwise convergence: Pointwise convergence refers to the process where a sequence of functions converges to a limit function at each individual point in their domain. This means that for every point in the domain, the values of the sequence of functions approach the value of the limit function as the index goes to infinity. It is important because it helps to understand how functions behave when approximated by sequences and is connected to many concepts in analysis.
Riemann Mapping Theorem: The Riemann Mapping Theorem states that any simply connected open subset of the complex plane, which is not the entire plane, can be conformally mapped to the open unit disk. This theorem is crucial for understanding how complex functions can transform regions in the plane, and it connects deeply with conformal mappings and their properties, particularly when examining how elementary functions behave on these domains.
Singularities: Singularities are points in the complex plane where a function ceases to be well-defined or fails to be analytic, leading to behaviors that can include poles, essential singularities, or removable singularities. Understanding singularities is crucial because they can dramatically influence the properties of functions, especially when examining mappings by elementary functions or when applying these concepts in practical scenarios such as physics and engineering.
Surjectivity: Surjectivity is a property of a function where every element in the target set has at least one pre-image in the domain set. This means that for a function to be surjective, every possible output value must be achievable by some input value from the domain. Surjectivity plays a crucial role in determining whether mappings by elementary functions cover their entire range, connecting to other important concepts such as injectivity and bijectivity.
Uniform Convergence: Uniform convergence is a type of convergence for sequences of functions where the speed of convergence is uniform across the entire domain. This means that for any chosen level of accuracy, there is a single point in the domain from which all functions converge uniformly, ensuring the limit function preserves continuity and other properties of the original functions.
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