Trigonometric Identities
Reduction
cos(2t)=1−2sin2(t)
\( \cos(2t)=2\cos^2(t)-1 \)
Square
\( \sin^2(t)=\frac{1-\cos(2t)}{2} \)
\( \cos(2t)=\cos^2(t)-\sin^2(t) \)
\( \cos^2(t)=\frac{1+\cos(2t)}{2} \)
\( \sin(2t)=2\sin(t)\cos(t) \)
Ακολουθίες
Για \( n \) βαθμού πολυώνυμο \( p(r) \) με ρίζα \( r_1 \) πολλαπλότητας \( m_1 \), ισχύει: \( p'(r_1) = 0 \), ..., \( p^{(m_1-1)}(r_1) = 0 \), αφού: \( p^{k}(r_1) \varpropto (r - r_1)^{m-k} \).
Linear Algebra
Derivatives
\( \left. \frac{\partial W(x, \; ...)}{\partial x} \right|_{\text{varying x, constant everything else}} = \left. \frac{\partial W(r(x, \; ...), \; ...)}{\partial r} \right|_{\text{varying r(x), constant everything else}} \left. \frac{\partial r(x, \; ...) }{\partial x} \right|_{\text{varying x, constant everything else}} \)
Οι μιγαδικές λύσεις δευτεροβάθμιας είναι συζυγείς.
\( \frac{\partial}{\partial x} a^{x} = \frac{\partial}{\partial x} e^{\ln{a^{x}}} = \frac{\partial}{\partial x} e^{x \ln{a}} = \ln{a} \; e^{x \ln{a}} = a^x \; \ln{a} \)
Linear Algebra
Determinants
Convex Function (Bowl-shaped)
Univariate = Of one variable
Binomial Theorem:
\( (x + y)^n = \sum^n_{k = 0} \binom{n}{k} x^{n-k} y^k = \sum^n_{k = 0} \binom{n}{k} x^k y^{n-k} \)