A complex number \( z \) is defined as:
\[ z = a + bi \]
where:
Complex numbers are represented in the complex plane (Argand diagram), where:
Euler's formula establishes a deep connection between exponential and trigonometric functions:
\[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \]
A complex number \( z = a + bi \) can be expressed in polar form as:
\[ z = r(\cos\theta + i\sin\theta) = re^{i\theta} \]
where:
De Moivre's theorem states:
\[ (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) \]
for any integer \( n \) and real \( \theta \).
The \( n \)-th roots of a complex number \( z = r(\cos\theta + i\sin\theta) \) are given by:
\[ z_k = r^{1/n} \left[ \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right] \]
for \( k = 0, 1, 2, \dots, n-1 \).
Find the cube roots of \( z = 8(\cos(\pi/3) + i\sin(\pi/3)) \):
A function of a complex variable is a rule that assigns a complex number \( w \) to each complex number \( z \) in a domain \( D \subseteq \mathbb{C} \).
\[ f(z) = a_n z^n + a_{n-1} z^{n-1} + \dots + a_0 \]
\[ f(z) = \frac{P(z)}{Q(z)} \]
where \( P \) and \( Q \) are polynomials.\[ f(z) = e^z = e^x (\cos y + i\sin y) \]
where \( z = x + iy \).\[ f(z) = \ln|z| + i\arg(z) \]
A function \( f(z) \) is analytic (holomorphic) at a point \( z_0 \) if it is differentiable in some neighborhood of \( z_0 \).
For \( f(z) = u(x, y) + iv(x, y) \) to be analytic, it must satisfy:
Additionally, \( u \) and \( v \) must satisfy the Laplace equation:
A point \( z_0 \) is a singularity of \( f(z) \) if \( f(z) \) is not analytic at \( z_0 \) but is analytic in some punctured neighborhood of \( z_0 \).
A branch point is a point \( z_0 \) such that a function \( f(z) \) is not single-valued in any neighborhood of \( z_0 \). Common in multi-valued functions like \( \log(z) \) and \( z^{1/n} \).
The order of a singularity at \( z_0 \) is determined by the Laurent series expansion of \( f(z) \) around \( z_0 \):
A branch cut is a curve in the complex plane that restricts the domain of a multi-valued function to make it single-valued and continuous.
For the function \( f(z) = \log(z) \), a branch cut is typically made along the negative real axis (from \( 0 \) to \( -\infty \)).
The integral of a complex function \( f(z) \) along a contour \( C \) is defined as:
\[ \int_C f(z) \, dz = \int_a^b f(\gamma(t)) \, \gamma'(t) \, dt \]
where \( \gamma(t) \) is a parameterization of \( C \) for \( t \in [a, b] \).
\[ \int_C f(z) \, dz = 0 \]
for any closed contour \( C \) in \( D \).Cauchy's inequality provides a bound on the coefficients of a Taylor series for an analytic function. If \( f(z) \) is analytic in a disk \( |z - z_0| < R \), then:
\[ |a_n| \leq \frac{M}{R^n} \]
where:
If \( f(z) \) is analytic in a simply connected domain \( D \), and \( C \) is a simple closed contour in \( D \) that encloses \( z_0 \), then:
\[ f(z_0) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_0} \, dz \]
For the \( n \)-th derivative of \( f(z) \) at \( z_0 \):
\[ f^{(n)}(z_0) = \frac{n!}{2\pi i} \int_C \frac{f(z)}{(z - z_0)^{n+1}} \, dz \]
This documentation provides a overview of Complex Analysis, with LaTeX-formatted equations and SVG graphs for clarity. Each section can be expanded with additional examples, proofs, or visualizations as needed.