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--- math [2025-11-14 Fri 00:24] – created theorytoe
+++ math [2025-12-14 Sun 02:19] (current) – theorytoe
@@ Line 1: / Line 1: @@
-[[math:transforms|List of Integral / Space Transforms]]
+====== Transform Tables ======
+  * [[math:transforms|List of Integral / Space Transforms]]
+  * [[math:fourier|Fourier Analysis]]
+  * [[math:units|Units and System Conversions]]
+====== Useful Equations ======
+^ Eulers Formula    | $$e^{j\varphi} = \cos(\varphi)+j\sin(\varphi)$$      |
+^ Eulers Cosine     | $$\cos(x) = \frac{e^{jx}}{2} + \frac{e^{-jx}}{2}$$   |
+^ Eulers Sine       | $$\sin(x) = \frac{je^{-jx}}{2} - \frac{je^{jx}}{2}$$ |
+^ Angular Frequency | $$\omega = 2\pi f$$                                  |
+^ Period            | $$T = \frac{1}{f}$$                                  |
+====== Common Integrals ======
+| $$\int\,u\,dv = uv-\int\,v\,du$$                   |
+| $$\int\,\frac{1}{ax+b}\,dx = \frac{1}{a}ln(ax+b)$$ |
+| $$\int\,\sin(x)\,dx = -\cos(x)$$                   |
+| $$\int\,\cos(x)\,dx = \sin(x)$$                    |
+| $$\int\,e^{ax}\,dx = \frac{1}{a}e^{ax}$$           |
+| $$\int\,xe^x\,dx = (x-1)e^x$$                      |
+====== Common Derivatives ======
+| $$\frac{d}{dx}(f(x)g(x)) = f(x)\dot{g}(x) + \dot{f}(x)g(x)$$                                    |
+| $$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f(x)\dot{g}(x)-\dot{f}(x)g(x)}{(g(x))^2}$$ |
+| $$\frac{d}{dx}(\sin(x)) = \cos(x) $$                                                            |
+| $$\frac{d}{dx}(\cos(x)) = -\sin(x) $$                                                           |
+| $$\frac{d}{dx}(\tan(x)) = \sec^2(x) $$                                                          |
+| $$\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$$                                         |
+| $$\frac{d}{dx}(\cos^{-1}(x)) = \frac{-1}{\sqrt{1-x^2}}$$                                        |
+| $$\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2}$$                                                |
+| $$\frac{d}{dx}(a^x) = a^x\ln(a)$$                                                               |
+| $$\frac{d}{dx}(\ln|x|) = \frac{1}{x}$$                                                          |
+| $$\frac{d}{dx}(\log_a(x)) = \frac{1}{x\ln(a)}$$                                                 |

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