Formulas

Author

Mohammad Alkousa

Published

September 26, 2025

Complex Numbers

Imaginary Unit: \(i^2 = -1\)
Standard Form of a Complex Number: \(z = a + bi\)
Modulus of a Complex Number: \(|z| = \sqrt{a^2 + b^2}\)
Argument of a Complex Number: \(\theta = \arctan(b/a)\)
Equation of a Circle in the Complex Plane: \(|z - c| = r\)
Trigonometric Form: \(z = r(\cos\theta + i\sin\theta)\)
Euler’s Formula: \(e^{i\theta} = \cos\theta + i\sin\theta\)
Exponential (Polar) Form: \(z = re^{i\theta}\)
System for Square Roots of \(x+yi\): \[ \begin{cases} a^2 - b^2 = x \\ 2ab = y \end{cases} \]
De Moivre’s Theorem (Powers): \(z^n = r^n(\cos(n\theta) + i\sin(n\theta))\)
De Moivre’s Theorem (Roots): \[ \sqrt[n]{z} = \sqrt[n]{r} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \] for \(k = 0, 1, \dots, n-1\).

Functions

Even Part of a Function: \[f_{\text{even}}(x) = \frac{f(x) + f(-x)}{2}\]
Odd Part of a Function: \[f_{\text{odd}}(x) = \frac{f(x) - f(-x)}{2}\]
Function Composition: \[(f \circ g)(x) = f(g(x))\]
Finding an Inverse Function: \(y=f(x) \implies x=f^{-1}(y)\)

Trigonometry

Pythagorean Identity: \[\sin^2 x + \cos^2 x = 1\]
Inverse Trigonometric Identity: \[\arcsin(x) + \arccos(x) = \frac{\pi}{2}\]
Inverse Sine as an Odd Function: \[\arcsin(-x) = -\arcsin(x)\]
Inverse Reciprocal Identity: \[\arcsin\left(\frac{1}{x}\right) = \text{arccsc}(x)\]

Hyperbolic Functions

Hyperbolic Sine Definition: \[\sinh x = \frac{e^x - e^{-x}}{2}\]
Hyperbolic Cosine Definition: \[\cosh x = \frac{e^x + e^{-x}}{2}\]
Hyperbolic Identity: \[\cosh^2 x - \sinh^2 x = 1\]
Hyperbolic-Inverse Trig Identities:
- \(\sinh(\text{arccosh } x) = \sqrt{x^2 - 1}\)
- \(\sinh(\text{arctanh } x) = \frac{x}{\sqrt{1-x^2}}\)
- \(\cosh(\text{arcsinh } x) = \sqrt{1+x^2}\)
- \(\cosh(\text{arctanh } x) = \frac{1}{\sqrt{1-x^2}}\)
- \(\tanh(\text{arcsinh } x) = \frac{x}{\sqrt{1+x^2}}\)
- \(\tanh(\text{arccosh } x) = \frac{\sqrt{x^2-1}}{x}\)
Logarithmic Form of arsinh: \[\text{arsinh}(x) = \ln(x + \sqrt{x^2+1})\]
Logarithmic Form of arcosh: \[\text{arcosh}(x) = \ln(x + \sqrt{x^2-1})\]
Logarithmic Form of artanh: \[\text{artanh}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)\]
Logarithmic Form of arcsch: \[\text{arcsch}(x) = \ln\left(\frac{1}{x} + \frac{\sqrt{1+x^2}}{|x|}\right)\]
Logarithmic Form of arsech: \[\text{arsech}(x) = \ln\left(\frac{1 + \sqrt{1-x^2}}{x}\right)\]
Logarithmic Form of arcoth: \[\text{arcoth}(x) = \frac{1}{2}\ln\left(\frac{x+1}{x-1}\right)\]

Algebra and Inequalities

Binomial Theorem: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
Sum of Cubes: \[1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}\]
Triangle Inequality: \(|x + y| \le |x| + |y|\)
Bernoulli’s Inequality: For \(x > -1\) and integer \(n \ge 1\), \((1+x)^n \ge 1+nx\).

Sequences and Limits

Arithmetic Sequence (n-th term): \(a_n = a_1 + (n-1)r\)
Arithmetic Sequence (Sum): \[S_n = \frac{n}{2}(a_1 + a_n)\]
Geometric Sequence (n-th term): \(a_n = a_1 q^{n-1}\)
Geometric Sequence (Sum): \[S_n = a_1 \frac{1-q^n}{1-q}\]
Limit Definition: \(\lim_{n \to \infty} a_n = L \iff (\forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } n > N \implies |a_n - L| < \epsilon)\)
Divergence to Infinity Definition: \(\lim_{n \to \infty} a_n = \infty \iff (\forall M > 0, \exists N \in \mathbb{N} \text{ such that } n > N \implies a_n > M)\)
Squeeze Theorem: If \(a_n \le b_n \le c_n\) for \(n>N\) and \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L\), then \(\lim_{n \to \infty} b_n = L\).
Sum Rule for Limits: \(\lim_{n \to \infty} (a_n + b_n) = \lim_{n \to \infty} a_n + \lim_{n \to \infty} b_n\)
Constant Multiple Rule for Limits: \(\lim_{n \to \infty} (\alpha a_n) = \alpha \lim_{n \to \infty} a_n\)
Product Rule for Limits: \(\lim_{n \to \infty} (a_n \cdot b_n) = (\lim_{n \to \infty} a_n) \cdot (\lim_{n \to \infty} b_n)\)
Quotient Rule for Limits: \[\lim_{n \to \infty} \left(\frac{a_n}{b_n}\right) = \frac{\lim_{n \to \infty} a_n}{\lim_{n \to \infty} b_n}\]
The Number e: \[e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n\]
Limit Definition of e: \(\lim_{n\to\infty} \left(1 + \frac{x}{n}\right)^n = e^x\)
Ratio Test for Sequences: Given \(L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|\), if \(L < 1\) then \(\lim_{n \to \infty} a_n = 0\).
Cauchy Sequence Definition: For every \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that for any \(m, n > N\), we have \(|a_m - a_n| < \epsilon\).
Recursive Limit Evaluation: If \(\lim_{n \to \infty} a_n = L\) and \(a_{n+1} = f(a_n)\) where \(f\) is continuous, then \(L=f(L)\).

Series

Series Notation: \(\sum_{n=1}^{\infty} a_n = a_1 + a_2 + \dots + a_n + \dots\)
nth Partial Sum: \(s_n = \sum_{i=1}^{n} a_i = a_1 + a_2 + \dots + a_n\)
Sum of a Convergent Series: \(s = \lim_{n \to \infty} s_n\)
General Telescoping Series Sum: If \(a_n = b_n - b_{n+1}\), then \(\sum_{n=1}^{\infty} a_n = b_1 - \lim_{n \to \infty} b_{n+1}\).
Mengoli Series: \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)} = 1\)
Generalized Telescoping Sum: \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)\cdots(n+r)} = \frac{1}{r \cdot r!}\)
Geometric Series Sum: For \(|q| < 1\), \(\sum_{n=0}^{\infty} aq^n = \frac{a}{1-q}\).
Harmonic Series Divergence: \(\sum_{n=1}^{\infty} \frac{1}{n} = \infty\)
\(p\)-Series Convergence: \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converges if and only if \(p > 1\).
Linear Property of Series: \(\sum_{n=1}^{\infty}(\alpha a_n + \beta b_n) = \alpha \sum_{n=1}^{\infty} a_n + \beta \sum_{n=1}^{\infty} b_n\)
Necessary Condition for Convergence: If \(\sum a_n\) converges, then \(\lim_{n \to \infty} a_n = 0\).
nth-Term Test for Divergence: If \(\lim_{n \to \infty} a_n \neq 0\) or the limit does not exist, then \(\sum a_n\) diverges.
Direct Comparison Test: If \(0 \leq a_n \leq b_n\), then: (1) \(\sum b_n\) converges \(\Rightarrow\) \(\sum a_n\) converges; (2) \(\sum a_n\) diverges \(\Rightarrow\) \(\sum b_n\) diverges.
Limit Comparison Test: If \(\lim_{n \to \infty} \frac{a_n}{b_n} = L\) where \(0 < L < \infty\), then \(\sum a_n\) and \(\sum b_n\) have the same behavior.
Ratio Test: Let \(L = \lim_{n\to\infty} |\frac{a_{n+1}}{a_n}|\). If \(L < 1\), the series converges absolutely. If \(L > 1\), the series diverges. If \(L = 1\), inconclusive.
Root Test: Let \(L = \lim_{n\to\infty} \sqrt[n]{|a_n|}\). If \(L < 1\), the series converges absolutely. If \(L > 1\), the series diverges. If \(L = 1\), inconclusive.
Leibniz Test: If \(\{a_n\}\) is non-increasing and \(\lim a_n = 0\), then \(\sum (-1)^{n+1} a_n\) converges.
Absolute Convergence Test: If \(\sum |a_n|\) converges, then \(\sum a_n\) converges.
Alternating Harmonic Series Sum: \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \ln(2)\)
Cauchy Product: \((\sum a_n)(\sum b_n) = \sum c_n\) where \(c_n = \sum_{k=0}^{n} a_k b_{n-k}\)

Power Series

Power Series Form: \(\sum_{n=0}^{\infty} a_n(x-c)^n\)
Radius of Convergence (Ratio Test): \(R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right|\)
Radius of Convergence (Root Test): \(R = \frac{1}{\lim_{n \to \infty} \sqrt[n]{|a_n|}}\)

Function Limits and Continuity

Limit Definition (\(\varepsilon\)-\(\delta\)): \(\lim_{x \to c} f(x) = L \iff (\forall \varepsilon > 0, \exists \delta > 0: 0 < |x-c| < \delta \implies |f(x)-L| < \varepsilon)\)
Right-Hand Limit: \(\lim_{x \to c^+} f(x) = L \iff (\forall \varepsilon > 0, \exists \delta > 0: c < x < c+\delta \implies |f(x)-L| < \varepsilon)\)
Left-Hand Limit: \(\lim_{x \to c^-} f(x) = L \iff (\forall \varepsilon > 0, \exists \delta > 0: c-\delta < x < c \implies |f(x)-L| < \varepsilon)\)
Limit Existence: \(\lim_{x \to c} f(x) = L \iff \lim_{x \to c^-} f(x) = L \text{ and } \lim_{x \to c^+} f(x) = L\)
Limit at Infinity: \(\lim_{x \to \infty} f(x) = L \iff (\forall \varepsilon > 0, \exists M: x > M \implies |f(x)-L| < \varepsilon)\)
Infinite Limit: \(\lim_{x \to c} f(x) = \infty \iff (\forall B > 0, \exists \delta > 0: 0 < |x-c| < \delta \implies f(x) > B)\)
Sum Rule for Function Limits: \(\lim_{x \to c} (f(x) + g(x)) = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)\)
Product Rule for Function Limits: \(\lim_{x \to c} (f(x) \cdot g(x)) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)\)
Quotient Rule for Function Limits: \(\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}\) (if denominator \(\neq 0\))
Power Rule for Function Limits: \(\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n\)
Root Rule for Function Limits: \(\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)}\) (with domain restrictions)
Polynomial Limit: If \(P(x)\) is a polynomial, then \(\lim_{x \to c} P(x) = P(c)\)
Rational Function Limit: \(\lim_{x \to c} \frac{P(x)}{Q(x)} = \frac{P(c)}{Q(c)}\) if \(Q(c) \neq 0\)
Sandwich (Squeeze) Theorem: If \(g(x) \le f(x) \le h(x)\) near \(c\) and \(\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L\), then \(\lim_{x \to c} f(x) = L\)
Important Trigonometric Limit: \(\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\) (\(\theta\) in radians)
Related Trigonometric Limits: \(\lim_{\theta \to 0} \sin \theta = 0\), \(\lim_{\theta \to 0} \cos \theta = 1\)
Tangent Limit: \(\lim_{\theta \to 0} \frac{\tan \theta}{\theta} = 1\)
Exponential Limit (Form 1): \(\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e\)
Exponential Limit (Form 2): \(\lim_{x \to 0} (1 + x)^{1/x} = e\)
General Exponential Limit: \(\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x = e^a\)
Exponential-Logarithmic Limit: \(\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a\) (for \(a > 0\))
Indeterminate Form \(1^{\infty}\): If \(\lim_{x \to c} g(x) = 1\) and \(\lim_{x \to c} h(x) = \infty\), then \(\lim_{x \to c} (g(x))^{h(x)} = e^{\lim_{x \to c} [h(x)(g(x)-1)]}\)
Continuity at a Point: \(f\) is continuous at \(c\) if \(\lim_{x \to c} f(x) = f(c)\)
Composition of Continuous Functions: If \(f\) is continuous at \(c\) and \(g\) is continuous at \(f(c)\), then \(g \circ f\) is continuous at \(c\)
Limit of Continuous Composition: If \(\lim_{x \to c} f(x) = b\) and \(g\) is continuous at \(b\), then \(\lim_{x \to c} g(f(x)) = g(b)\)
Intermediate Value Theorem: If \(f\) is continuous on \([a,b]\) and \(y_0\) is between \(f(a)\) and \(f(b)\), then \(\exists c \in [a,b]\) such that \(f(c) = y_0\)

Differentiation

Basic Differentiation Rules:

Constant Rule: \(\frac{d}{dx}(c) = 0\)
Power Rule: \(\frac{d}{dx}(x^n) = nx^{n-1}\) for any real \(n\)
Constant Multiple Rule: \(\frac{d}{dx}(cu) = c\frac{du}{dx}\)
Sum Rule: \(\frac{d}{dx}(u + v) = \frac{du}{dx} + \frac{dv}{dx}\)
Difference Rule: \(\frac{d}{dx}(u - v) = \frac{du}{dx} - \frac{dv}{dx}\)
Product Rule: \(\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}\)
Quotient Rule: \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\) (where \(v \neq 0\))
Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)

Derivatives of Elementary Functions:

Exponential Functions:
- \(\frac{d}{dx}(e^x) = e^x\)
- \(\frac{d}{dx}(a^x) = a^x \ln a\) for \(a > 0\)
Logarithmic Functions:
- \(\frac{d}{dx}(\ln x) = \frac{1}{x}\) for \(x > 0\)
- \(\frac{d}{dx}(\ln |x|) = \frac{1}{x}\) for \(x \neq 0\)
Trigonometric Functions:
- \(\frac{d}{dx}(\sin x) = \cos x\)
- \(\frac{d}{dx}(\cos x) = -\sin x\)
- \(\frac{d}{dx}(\tan x) = \sec^2 x = \frac{1}{\cos^2 x}\)
- \(\frac{d}{dx}(\cot x) = -\csc^2 x = -\frac{1}{\sin^2 x}\)
- \(\frac{d}{dx}(\sec x) = \sec x \tan x\)
- \(\frac{d}{dx}(\csc x) = -\csc x \cot x\)
Inverse Trigonometric Functions:
- \(\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}\) for \(|x| < 1\)
- \(\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}\) for \(|x| < 1\)
- \(\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}\) for all \(x\)
- \(\frac{d}{dx}(\text{arccot } x) = -\frac{1}{1+x^2}\)
- \(\frac{d}{dx}(\text{arcsec } x) = \frac{1}{|x|\sqrt{x^2-1}}\) for \(|x| > 1\)
- \(\frac{d}{dx}(\text{arccsc } x) = -\frac{1}{|x|\sqrt{x^2-1}}\) for \(|x| > 1\)
Hyperbolic Functions:
- \(\frac{d}{dx}(\sinh x) = \cosh x\)
- \(\frac{d}{dx}(\cosh x) = \sinh x\)
- \(\frac{d}{dx}(\tanh x) = \frac{1}{\cosh^2 x}\)

Chain Rule Forms (where \(u = u(x)\)):

\((u^n)' = nu^{n-1}u'\)
\((e^u)' = e^u u'\)
\((a^u)' = a^u u' \ln a\) for \(a > 0\)
\((\ln u)' = \frac{u'}{u}\) for \(u > 0\)
\((\sin u)' = u' \cos u\)
\((\cos u)' = -u' \sin u\)
\((\tan u)' = \frac{u'}{\cos^2 u}\)
\((\arcsin u)' = \frac{u'}{\sqrt{1-u^2}}\) for \(|u| < 1\)
\((\arctan u)' = \frac{u'}{1+u^2}\)

Inverse Function Rule:

\((f^{-1})'(b) = \frac{1}{f'(f^{-1}(b))}\) where \(b = f(a)\)

Higher-Order Derivatives:

Leibniz Formula: \((fg)^{(n)} = \sum_{k=0}^n \binom{n}{k} f^{(n-k)} g^{(k)}\)
Common Higher Derivatives:
- \((\ln x)^{(n)} = \frac{(-1)^{n-1}(n-1)!}{x^n}\) for \(n \ge 1\)
- \((\sin x)^{(n)} = \sin(x + \frac{n\pi}{2})\) for \(n \ge 1\)
- \((\cos x)^{(n)} = \cos(x + \frac{n\pi}{2})\) for \(n \ge 1\)
- \((x^{-1})^{(n)} = \frac{(-1)^n n!}{x^{n+1}}\) for \(n \ge 1\)

L’Hôpital’s Rule:

If \(\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0\) or both \(= \pm\infty\), then: \[\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\] (provided the right side exists or is \(\pm\infty\))

Taylor Series (Maclaurin Series at \(x = 0\)):

General Formula: \(f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(c)}{n!}(x-c)^n\)
Taylor’s Formula: \(f(x) = T_n(x) + R_n(x)\) where \(R_n(x) = \frac{f^{(n+1)}(\alpha)}{(n+1)!}(x-c)^{n+1}\)
Common Series (at \(x = 0\)):
- \(e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\)
- \(\sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\)
- \(\cos x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\)
- \(\ln(1+x) = \sum_{n=1}^\infty (-1)^{n+1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\) for \(|x| < 1\)
- \((1+x)^\alpha = \sum_{n=0}^\infty \binom{\alpha}{n} x^n = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \cdots\) for \(|x| < 1\)
- \(\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \cdots\) for \(|x| < 1\)
- \(\arctan x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots\) for \(|x| \le 1\)

Theorems and Tests:

Fermat’s Theorem: If \(f\) has a local extremum at interior point \(c\) and \(f'(c)\) exists, then \(f'(c) = 0\)
Rolle’s Theorem: If \(f\) is continuous on \([a,b]\), differentiable on \((a,b)\), and \(f(a) = f(b)\), then \(\exists c \in (a,b)\) with \(f'(c) = 0\)
Mean Value Theorem: If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then \(\exists c \in (a,b)\) with: \[f'(c) = \frac{f(b)-f(a)}{b-a}\]
Monotonicity Test:
- \(f'(x) > 0\) on \((a,b) \implies f\) increasing on \([a,b]\)
- \(f'(x) < 0\) on \((a,b) \implies f\) decreasing on \([a,b]\)
First Derivative Test: At critical point \(c\):
- \(f'\) changes from \(-\) to \(+\): local minimum
- \(f'\) changes from \(+\) to \(-\): local maximum
- \(f'\) doesn’t change sign: no extremum
Second Derivative Test: If \(f'(c) = 0\):
- \(f''(c) > 0\): local minimum
- \(f''(c) < 0\): local maximum
- \(f''(c) = 0\): inconclusive
Concavity Test:
- \(f''(x) > 0\) on \(I \implies f\) concave up on \(I\)
- \(f''(x) < 0\) on \(I \implies f\) concave down on \(I\)