math117: add up to pfd

docs/ce1/math117.md

@@ -43,7 +43,7 @@ If a function is not invertible, restricting the domain may allow a **partial in
     By restricting the domain to $[0,\inf]$, the **multivalued inverse function** $y=\pm\sqrt{x}$ is reduced to just the partial inverse $y=\sqrt{x}$.
 
 ## Symmetry
-
+78u7u887878
 An **even function** satisfies the property that $f(x)=f(-x)$, indicating that it is unchanged by a reflection across the y-axis.
 
 An **odd function** satisfies the property that $-f(x)=f(-x)$, indicating that it is unchanged by a 180° rotation about the origin.
@@ -70,3 +70,133 @@ $$
     \sinh x = \frac{1}{2}(e^x - e^{-x})
     $$
 
+## Piecewise functions
+
+A piecewise function is one that changes formulae at certain intervals. To solve piecewise functions, each of one's intervals should be considered.
+
+### Absolute value function
+
+$$
+\begin{align*}
+|x| = \begin{cases}
+x &\text{ if } x\geq 0 \\
+-x &\text{ if } x < 0
+\end{cases}
+\end{align*}
+$$
+
+### Signum function
+
+The signum function returns the sign of its argument.
+
+$$
+\begin{align*}
+\text{sgn}(x)=\begin{cases}
+-1 &\text{ if } x < 0 \\
+0 &\text{ if } x = 0 \\
+1 &\text{ if } x > 0
+\end{cases}
+\end{align*}
+$$
+
+### Ramp function
+
+The ramp function makes a ramp through the origin that suddenly flatlines at 0. Where $c$ is a constant:
+
+$$
+\begin{align*}
+r(t)=\begin{cases}
+0 &\text{ if } x \leq 0 \\
+ct &\text{ if } x > 0
+\end{cases}
+\end{align*}
+$$
+
+<img src="https://upload.wikimedia.org/wikipedia/commons/c/c9/Ramp_function.svg" width=700>(Source: Wikimedia Commons, public domain)</img>
+
+### Floor and ceiling functions
+
+The floor function rounds down.
+$$\lfloor x\rfloor$$
+
+The ceiling function rounds up.
+$$\lceil x \rceil$$
+
+### Fractional part function
+
+In a nutshell, the fractional part function:
+
+ - returns the part **after the decimal point** if the number is positive
+ - returns 1 - **the part after the decimal point** if the number is negative
+
+$$\text{FRACPT}(x) = x-\lfloor x\rfloor$$
+
+Because this function is periodic, it can be used to limit angles to the $[0, 2\pi)$ range with:
+$$f(\theta) = 2\pi\cdot\text{FRACPT}\biggr(\frac{\theta}{2\pi}\biggr)$$
+
+### Heaviside function
+
+The Heaviside function effectively returns a boolean whether the number is greater than 0.
+$$
+\begin{align*}
+H(x) = \begin{cases}
+0 &\text{ if } t < 0 \\
+1 &\text{ if } t \geq 0
+\end{cases}
+\end{align*}
+$$
+
+This can be used to construct other piecewise functions by enabling them with $H(x-a)$ as a factor, where $a$ is the interval.
+
+In a nutshell:
+
+ - $1-H(t-a)$ lets you "turn a function off" at at $t=a$
+ - $H(t-a)$ lets you "turn a function on at $t=a$
+ - $H(t-a) - H(t-b)$ leaves a function on in the interval $(a, b)$
+
+!!! example
+    TODO: example for converting piecewise to heaviside via collecting heavisides
+    
+    and vice versa
+    
+## Periodicity
+
+The function $f(t)$ is periodic only if there is a repeating pattern, i.e. such that for every $x$, there is an $f(x) = f(x + nT)$, where $T$ is the period and $n$ is any integer.
+
+### Circular motion
+
+Please see [SL Physics 1#6.1 - Circular motion](/g11/sph3u7/#61-circular-motion) and its subcategory "Angular thingies" for more information.
+
+## Partial function decomposition (PFD)
+
+In order to PFD:
+
+1. Factor the denominator into irreducibly quadratic or linear terms. 
+2. For each factor, create a term. Where capital letters below are constants:
+  - A linear factor $Bx+C$ has a term $\frac{A}{Bx+C}$.
+  - A quadratic factor $Dx^2+Ex+G$ has a term $\frac{H}{Dx^2+Ex+G}$.
+3. Set the two equal to each other such that the denominators can be factored out.
+4. Create systems of equations to solve for each constant.
+
+!!! example
+    To decompose $\frac{x}{(x+1)(x^2+x+1)}$:
+    $$
+    \begin{align*}
+    \frac{x}{(x+1)(x^2+x+1)} &= \frac{A}{x+1} + \frac{Bx+C}{x^2+x+1} \\
+    &= \frac{A(x^2+x+1) + (Bx+C)(x+1)}{(x+1)(x^2+x+1)} \\
+    x &= A(x^2+x+1) + (Bx+C)(x+1) \\
+    0x^2 + x + 0 &= (Ax^2 + Bx^2) + (Ax + Bx + Cx) + (A + C) \\
+    \\
+    &\begin{cases}
+    0 = A + B \\
+    1 = A + B + C \\
+    0 = A + C
+    \end{cases}
+    \\
+    A &= -1 \\
+    B &= 1 \\
+    C &= 1 \\
+    \\
+    ∴ \frac{x}{(x+1)(x^2+x+1)} &= -\frac{1}{x+1} + \frac{x + 1}{x^2 + x + 1}
+    \end{align*}
+    $$