Тригонометрическая часть
[src] // 0 for And(im(x) = 0, x mod pi = 0)\ // 1 for And(im(x) = 0, x mod 2*pi = 0)\
2*|< |*|< |
\\sin(x) otherwise / \\cos(x) otherwise /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for And(im(x) = 0, x mod pi = 0)\ // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| | || |
|| /x\ | || 2/x\ |
|| 2*cot|-| | ||-1 + cot |-| |
2*|< \2/ |*|< \2/ |
||----------- otherwise | ||------------ otherwise |
|| 2/x\ | || 2/x\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
2
------------------
/pi \
csc(x)*csc|-- - x|
\2 /
$$\frac{2}{\csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
// 0 for And(im(x) = 0, x mod pi = 0)\
|| | // 1 for And(im(x) = 0, x mod 2*pi = 0)\
2*|< / pi\ |*|< |
||cos|x - --| otherwise | \\cos(x) otherwise /
\\ \ 2 / /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
/ 0 for And(im(x) = 0, 2*x mod pi = 0)
<
\sin(2*x) otherwise
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}$$
// 1 for And(im(x) = 0, x mod 2*pi = 0)\
// 0 for And(im(x) = 0, x mod pi = 0)\ || |
|| | || 1 |
2*|< 1 |*|<----------- otherwise |
||------ otherwise | || /pi \ |
\\csc(x) / ||csc|-- - x| |
\\ \2 / /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
2*tan(x)
-----------
2
1 + tan (x)
$$\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
// 0 for And(im(x) = 0, x mod pi = 0)\ // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| | || |
|| /x\ | || 2/x\ |
|| 2*tan|-| | ||1 - tan |-| |
2*|< \2/ |*|< \2/ |
||----------- otherwise | ||----------- otherwise |
|| 2/x\ | || 2/x\ |
||1 + tan |-| | ||1 + tan |-| |
\\ \2/ / \\ \2/ /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
$$\sin{\left(2 x \right)}$$
// 0 for And(im(x) = 0, x mod pi = 0)\
|| | // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| 1 | || |
2*|<----------- otherwise |*|< 1 |
|| / pi\ | ||------ otherwise |
||sec|x - --| | \\sec(x) /
\\ \ 2 / /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)$$
2
------------------
/ pi\
sec(x)*sec|x - --|
\ 2 /
$$\frac{2}{\sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
// 1 for And(im(x) = 0, x mod 2*pi = 0)\
// 0 for And(im(x) = 0, x mod pi = 0)\ || |
2*|< |*|< / pi\ |
\\sin(x) otherwise / ||sin|x + --| otherwise |
\\ \ 2 / /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\sin{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
2
-------------
csc(x)*sec(x)
$$\frac{2}{\csc{\left(x \right)} \sec{\left(x \right)}}$$
/ pi\
cos|2*x - --|
\ 2 /
$$\cos{\left(2 x - \frac{\pi}{2} \right)}$$
/ pi\
2*sin(x)*sin|x + --|
\ 2 /
$$2 \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}$$
// 0 for And(im(x) = 0, x mod pi = 0)\ // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| | || |
2*| 0 for And(im(x) = 0, x mod pi = 0) |*| 1 for And(im(x) = 0, x mod 2*pi = 0) |
||< otherwise | ||< otherwise |
\\\sin(x) otherwise / \\\cos(x) otherwise /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 0 for And(im(x) = 0, x mod pi = 0)\ // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| | || |
||/ 0 for And(im(x) = 0, x mod pi = 0) | ||/ 1 for And(im(x) = 0, x mod 2*pi = 0) |
||| | ||| |
||| /x\ | ||| 2/x\ |
2*|<| 2*cot|-| |*|<|-1 + cot |-| |
||< \2/ otherwise | ||< \2/ otherwise |
|||----------- otherwise | |||------------ otherwise |
||| 2/x\ | ||| 2/x\ |
|||1 + cot |-| | |||1 + cot |-| |
\\\ \2/ / \\\ \2/ /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
1
-------------
/ pi\
sec|2*x - --|
\ 2 /
$$\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}}$$
/ 2/x\\ /x\
4*|1 - tan |-||*tan|-|
\ \2// \2/
----------------------
2
/ 2/x\\
|1 + tan |-||
\ \2//
$$\frac{4 \cdot \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
$$\frac{1}{\csc{\left(2 x \right)}}$$
/ pi\
2*cos(x)*cos|x - --|
\ 2 /
$$2 \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}$$
/ 0 for And(im(x) = 0, 2*x mod pi = 0)
|
| 2*cot(x)
<----------- otherwise
| 2
|1 + cot (x)
\
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}$$