📐Complex Analysis Unit 4 – Elementary Functions

Key Concepts and Definitions

• Complex numbers extend the real number system by introducing the imaginary unit $i$, defined as $i^2 = -1$
• A complex number $z$ is expressed as $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit
  • $a$ is called the real part of $z$, denoted as $\Re(z)$
  • $b$ is called the imaginary part of $z$, denoted as $\Im(z)$
• The complex conjugate of $z = a + bi$ is defined as $\bar{z} = a - bi$
• The modulus (or absolute value) of a complex number $z = a + bi$ is defined as $|z| = \sqrt{a^2 + b^2}$
• The argument (or phase) of a complex number $z = a + bi$ is defined as $\arg(z) = \arctan(\frac{b}{a})$, with appropriate adjustments based on the quadrant of $z$
• Euler's formula establishes the relationship between complex exponentials and trigonometric functions: $e^{i\theta} = \cos\theta + i\sin\theta$

Complex Numbers and the Complex Plane

• The complex plane (also known as the Argand plane) is a 2D representation of complex numbers
  • The horizontal axis represents the real part, and the vertical axis represents the imaginary part
• Complex numbers can be plotted on the complex plane, with the real part as the x-coordinate and the imaginary part as the y-coordinate
• The modulus of a complex number corresponds to the distance from the origin to the point representing the complex number on the complex plane
• The argument of a complex number is the angle between the positive real axis and the line segment connecting the origin to the point representing the complex number
• Complex numbers can be represented in various forms:
  • Rectangular (or Cartesian) form: $z = a + bi$
  • Polar form: $z = r(\cos\theta + i\sin\theta)$, where $r = |z|$ and $\theta = \arg(z)$
  • Exponential form: $z = re^{i\theta}$, derived from Euler's formula

Elementary Functions of Complex Variables

• Elementary functions of complex variables are extensions of real-valued functions to the complex domain
• The complex exponential function is defined as $e^z = e^{a+bi} = e^a(\cos b + i\sin b)$
• The complex logarithm function is the inverse of the complex exponential function and is defined as $\log z = \ln|z| + i\arg(z)$
  • The complex logarithm is multi-valued, with infinitely many branches separated by $2\pi i$
• Complex trigonometric functions (sine, cosine, tangent) are defined using Euler's formula:
  • $\sin z = \frac{e^{iz} - e^{-iz}}{2i}$
  • $\cos z = \frac{e^{iz} + e^{-iz}}{2}$
  • $\tan z = \frac{\sin z}{\cos z}$
• Complex hyperbolic functions (sinh, cosh, tanh) are defined analogously to their real-valued counterparts:
  • $\sinh z = \frac{e^z - e^{-z}}{2}$
  • $\cosh z = \frac{e^z + e^{-z}}{2}$
  • $\tanh z = \frac{\sinh z}{\cosh z}$

Properties and Behavior of Elementary Functions

• The complex exponential function is periodic with period $2\pi i$, i.e., $e^{z+2\pi i} = e^z$
• The complex logarithm function has a branch cut, typically chosen along the negative real axis
  • The principal branch of the logarithm is defined by restricting the imaginary part of the logarithm to the interval $(-\pi, \pi]$
• Complex trigonometric functions have periodicity similar to their real-valued counterparts:
  • $\sin(z + 2\pi) = \sin z$ and $\cos(z + 2\pi) = \cos z$
  • $\tan(z + \pi) = \tan z$
• Complex hyperbolic functions exhibit symmetry properties:
  • $\sinh(-z) = -\sinh z$ (odd function)
  • $\cosh(-z) = \cosh z$ (even function)
• The complex exponential, trigonometric, and hyperbolic functions satisfy many of the same identities as their real-valued counterparts, such as:
  • $\cosh^2 z - \sinh^2 z = 1$
  • $\cos^2 z + \sin^2 z = 1$

Analytic Properties and Singularities

• A complex function $f(z)$ is analytic (or holomorphic) at a point $z_0$ if it is differentiable in a neighborhood of $z_0$
  • Analyticity is a stronger condition than differentiability, as it requires the function to be differentiable in a region, not just at a single point
• The Cauchy-Riemann equations provide a necessary and sufficient condition for a complex function $f(z) = u(x, y) + iv(x, y)$ to be analytic:
  • $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• Elementary functions are analytic in their domains, except at certain points called singularities
• Types of singularities:
  • Removable singularities: The function can be redefined at the singular point to make it analytic (e.g., $\frac{\sin z}{z}$ at $z = 0$)
  • Poles: The function tends to infinity as $z$ approaches the singular point (e.g., $\frac{1}{z}$ at $z = 0$)
  • Essential singularities: The function exhibits complex behavior near the singular point (e.g., $e^{\frac{1}{z}}$ at $z = 0$)

Mapping and Transformations

• Complex functions can be visualized as mappings from one complex plane (the z-plane) to another complex plane (the w-plane)
• The mapping properties of elementary functions can be studied by examining how they transform certain regions or curves in the z-plane
• The complex exponential function $w = e^z$ maps:
  • Horizontal lines in the z-plane to circles centered at the origin in the w-plane
  • Vertical lines in the z-plane to rays emanating from the origin in the w-plane
• The complex logarithm function $w = \log z$ maps:
  • Circles centered at the origin in the z-plane to vertical lines in the w-plane
  • Rays emanating from the origin in the z-plane to horizontal lines in the w-plane
• Möbius transformations are a class of complex functions of the form $w = \frac{az + b}{cz + d}$, where $a, b, c, d$ are complex constants, and $ad - bc \neq 0$
  • Möbius transformations map circles and lines in the z-plane to circles and lines in the w-plane

Applications in Complex Analysis

• Complex analysis has numerous applications in various fields, including:
  • Physics: Quantum mechanics, electromagnetism, fluid dynamics
  • Engineering: Signal processing, control theory, electrical engineering
  • Mathematics: Number theory, algebraic geometry, differential equations
• The Fourier transform, which is widely used in signal processing, can be expressed using complex exponentials:
  • $\hat{f}(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t} dt$
• The Laplace transform, used in solving differential equations and analyzing control systems, involves complex exponentials:
  • $F(s) = \int_0^{\infty} f(t)e^{-st} dt$
• Complex analysis provides powerful tools for evaluating certain real integrals using techniques such as contour integration and the residue theorem

Common Pitfalls and Misconceptions

• Overextending properties of real-valued functions to complex functions without proper justification
  • For example, assuming that $\sqrt{z_1z_2} = \sqrt{z_1}\sqrt{z_2}$ for complex numbers $z_1$ and $z_2$, which is not always true
• Mishandling multi-valued functions, such as the complex logarithm or the complex square root
  • Failing to consider the principal branch or the choice of branch cut can lead to incorrect results
• Confusing the real and imaginary parts of a complex function with its components as a vector-valued function
• Misinterpreting the geometric meaning of certain complex operations, such as multiplication or division
  • Complex multiplication does not always correspond to a simple scaling and rotation as it does in the real case
• Neglecting the importance of domain and range when working with complex functions
  • The domain of a complex function may be restricted due to singularities or branch cuts, and the range may not cover the entire complex plane
• Misapplying analytic properties, such as assuming that a function is analytic just because it is differentiable at a point