Catalog / Complex Analysis Cheatsheet
Complex Analysis Cheatsheet
A comprehensive cheat sheet covering key concepts, theorems, and formulas in complex analysis, providing a quick reference for students and professionals.
Complex Numbers and Functions
Basic Definitions
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Complex Number: |
z = x + iy, where x and y are real numbers, and i is the imaginary unit (i² = -1). |
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Real Part: |
Re(z) = x |
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Imaginary Part: |
Im(z) = y |
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Complex Conjugate: |
\overline{z} = x - iy |
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Modulus: |
|z| = \sqrt{x^2 + y^2} |
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Argument: |
\theta = \arg(z), such that z = |z|e^{i\theta} |
Complex Functions
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Definition: |
A function f: \mathbb{C} \rightarrow \mathbb{C} that maps complex numbers to complex numbers. |
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Representation: |
f(z) = u(x, y) + iv(x, y), where u and v are real-valued functions. |
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Limit: |
\lim_{z \to z_0} f(z) = L if for every \epsilon > 0, there exists a \delta > 0 such that |f(z) - L| < \epsilon whenever 0 < |z - z_0| < \delta. |
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Continuity: |
f(z) is continuous at z_0 if \lim_{z \to z_0} f(z) = f(z_0). |
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Derivative: |
f'(z) = \lim_{h \to 0} \frac{f(z + h) - f(z)}{h}, if the limit exists. |
Polar Form
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Representation: |
z = r e^{i\theta} = r(\cos \theta + i \sin \theta), where r = |z| and \theta = \arg(z) |
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Multiplication: |
z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} |
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Division: |
\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} |
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Power: |
z^n = r^n e^{i n \theta} |
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Roots: |
z^{1/n} = r^{1/n} e^{i(\theta + 2\pi k)/n}, for k = 0, 1, ..., n-1 |
Analytic Functions
Cauchy-Riemann Equations
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Equations: |
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} |
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Analyticity: |
If f(z) = u(x, y) + iv(x, y) is analytic in a domain D, then the Cauchy-Riemann equations hold in D. |
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Sufficient Condition: |
If the partial derivatives of u and v are continuous and satisfy the Cauchy-Riemann equations in a domain D, then f(z) is analytic in D. |
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Complex Derivative: |
f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y} |
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Harmonic Functions: |
If f(z) is analytic, then u and v are harmonic functions, i.e., \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 and \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0. |
Elementary Functions
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Exponential Function: |
e^z = e^{x + iy} = e^x(\cos y + i \sin y) |
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Trigonometric Functions: |
\sin z = \frac{e^{iz} - e^{-iz}}{2i}, \cos z = \frac{e^{iz} + e^{-iz}}{2} |
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Hyperbolic Functions: |
\sinh z = \frac{e^z - e^{-z}}{2}, \cosh z = \frac{e^z + e^{-z}}{2} |
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Logarithmic Function: |
\log z = \ln |z| + i \arg z |
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Principal Value of Logarithm: |
\text{Log } z = \ln |z| + i \text{Arg } z, where -\pi < \text{Arg } z \leq \pi |
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Complex Power: |
z^c = e^{c \log z}, where c is a complex constant. |
Complex Integration
Contour Integrals
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Definition: |
\int_C f(z) dz = \int_a^b f(z(t)) z'(t) dt, where C is a smooth curve parameterized by z(t), a \leq t \leq b. |
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Properties: |
\int_C [\alpha f(z) + \beta g(z)] dz = \alpha \int_C f(z) dz + \beta \int_C g(z) dz |
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ML Estimate: |
|\int_C f(z) dz| \leq ML, where M is an upper bound for |f(z)| on C, and L is the length of C. |
Cauchy's Theorem and Integral Formula
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Cauchy’s Theorem: |
If f(z) is analytic in a simply connected domain D, then for any closed contour C in D, \oint_C f(z) dz = 0. |
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Cauchy’s Integral Formula: |
If f(z) is analytic in a simply connected domain D, and z_0 is any point in D inside a closed contour C, then f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - z_0} dz. |
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Generalized Integral Formula: |
f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z - z_0)^{n+1}} dz |
Series and Residues
Taylor and Laurent Series
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Taylor Series: |
If f(z) is analytic in a disk |z - z_0| < R, then f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n. |
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Laurent Series: |
If f(z) is analytic in an annulus r < |z - z_0| < R, then f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n, where a_n = \frac{1}{2\pi i} \oint_C \frac{f(z)}{(z - z_0)^{n+1}} dz. |
Residue Theorem
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Residue: |
The residue of f(z) at an isolated singularity z_0 is the coefficient a_{-1} in the Laurent series expansion of f(z) about z_0. |
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Residue Theorem: |
If f(z) is analytic inside and on a closed contour C, except for a finite number of isolated singularities z_k inside C, then \oint_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k). |
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Residue Calculation (Simple Pole): |
If f(z) has a simple pole at z_0, then \text{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z). |
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Residue Calculation (Pole of Order n): |
If f(z) has a pole of order n at z_0, then \text{Res}(f, z_0) = \frac{1}{(n-1)!} \lim_{z \to z_0} \frac{d^{n-1}}{dz^{n-1}} [(z - z_0)^n f(z)]. |
Applications of Residue Theorem
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Improper Integrals: |
The Residue Theorem can be used to evaluate improper integrals of the form \int_{-\infty}^{\infty} f(x) dx. |
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Trigonometric Integrals: |
The Residue Theorem can be used to evaluate integrals of the form \int_0^{2\pi} F(\cos \theta, \sin \theta) d\theta. |