Тригонометрическая часть
[src] 1
------------
6/pi \
csc |-- - x|
\2 / 1 csc 6 ( − x + π 2 ) \frac{1}{\csc^{6}{\left(- x + \frac{\pi}{2} \right)}} csc 6 ( − x + 2 π ) 1 // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| |
// 0 for And(im(x) = 0, x mod pi = 0)\ // 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)\ || 4 8/x\ |
|| | || | || | || | ||16*cos (x)*sin |-| |
- |< 6 | - |< 2 |*|< 2 | + |< 4/x\ 4/x\ | + |< \2/ |
||sin (x) otherwise | ||sin (x) otherwise | ||cos (x) otherwise | ||16*cos |-|*sin |-| otherwise | ||------------------ otherwise |
\\ / \\ / \\ / \\ \2/ \2/ / || 4 |
|| (-1 + cos(x)) |
\\ / ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 16 sin 4 ( x 2 ) cos 4 ( x 2 ) otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 2 ( x ) otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 cos 2 ( x ) otherwise ) ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 6 ( x ) otherwise ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 16 sin 8 ( x 2 ) cos 4 ( x ) ( cos ( x ) − 1 ) 4 otherwise ) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\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^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\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{16 \sin^{8}{\left(\frac{x}{2} \right)} \cos^{4}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{4}} & \text{otherwise} \end{cases}\right) ( { 0 16 sin 4 ( 2 x ) cos 4 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) − ( ( { 0 sin 2 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) ( { 1 cos 2 ( x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) ) − ( { 0 sin 6 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) + ⎩ ⎨ ⎧ 1 ( c o s ( x ) − 1 ) 4 16 s i n 8 ( 2 x ) c o s 4 ( x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise 6
/ 2/x\\
|1 - tan |-||
\ \2//
--------------
6
/ 2/x\\
|1 + tan |-||
\ \2// ( 1 − tan 2 ( x 2 ) ) 6 ( tan 2 ( x 2 ) + 1 ) 6 \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} ( tan 2 ( 2 x ) + 1 ) 6 ( 1 − tan 2 ( 2 x ) ) 6 // 1 for And(im(x) = 0, x mod 2*pi = 0)\
// 0 for And(im(x) = 0, x mod pi = 0)\ // 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)\ || |
|| | || | || | || | || 4 |
- |< 6 | - |< 2 |*|< 2 | + |< 4/x\ 8/x\ | + | ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 16 sin 8 ( x 2 ) cot 4 ( x 2 ) otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 2 ( x ) otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 cos 2 ( x ) otherwise ) ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 6 ( x ) otherwise ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( cot 2 ( x 2 ) − 1 ) 4 sin 8 ( x 2 ) otherwise ) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{x}{2} \right)} \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\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^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\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 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) ( { 0 16 sin 8 ( 2 x ) cot 4 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) − ( ( { 0 sin 2 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) ( { 1 cos 2 ( x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) ) − ( { 0 sin 6 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) + ( { 1 ( cot 2 ( 2 x ) − 1 ) 4 sin 8 ( 2 x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) 1 1 1 1
------- + ------------ - ------------ - --------------------
4 4/ pi\ 6/ pi\ 2 2/ pi\
sec (x) sec |x - --| sec |x - --| sec (x)*sec |x - --|
\ 2 / \ 2 / \ 2 / 1 sec 4 ( x − π 2 ) − 1 sec 6 ( x − π 2 ) − 1 sec 2 ( x ) sec 2 ( x − π 2 ) + 1 sec 4 ( x ) \frac{1}{\sec^{4}{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{6}{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec^{4}{\left(x \right)}} sec 4 ( x − 2 π ) 1 − sec 6 ( x − 2 π ) 1 − sec 2 ( x ) sec 2 ( x − 2 π ) 1 + sec 4 ( x ) 1 // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| |
|| 4 |
||/ 2/x\ \ |
// 0 for And(im(x) = 0, x mod pi = 0)\ // 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)\ ||| csc |-| | |
|| | || | || | || | ||| \2/ | |
|| 1 | || 1 | || 1 | || 16 | |||-1 + ------------| |
- |<------- otherwise | - |<------- otherwise |*|<------------ otherwise | + |<-------------------- otherwise | + |<| 2/pi x\| |
|| 6 | || 2 | || 2/pi \ | || 4/x\ 4/pi x\ | ||| csc |-- - -|| |
||csc (x) | ||csc (x) | ||csc |-- - x| | ||csc |-|*csc |-- - -| | ||\ \2 2// |
\\ / \\ / \\ \2 / / \\ \2/ \2 2/ / ||-------------------- otherwise |
|| 8/x\ |
|| csc |-| |
|| \2/ |
\\ / ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 16 csc 4 ( x 2 ) csc 4 ( − x 2 + π 2 ) otherwise ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 1 csc 6 ( x ) otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 1 csc 2 ( x ) otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 1 csc 2 ( − x + π 2 ) otherwise ) ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( csc 2 ( x 2 ) csc 2 ( − x 2 + π 2 ) − 1 ) 4 csc 8 ( x 2 ) otherwise ) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{16}{\csc^{4}{\left(\frac{x}{2} \right)} \csc^{4}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc^{6}{\left(x \right)}} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc^{2}{\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^{2}{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{4}}{\csc^{8}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) ( { 0 c s c 4 ( 2 x ) c s c 4 ( − 2 x + 2 π ) 16 for im ( x ) = 0 ∧ x mod π = 0 otherwise ) − ( { 0 c s c 6 ( x ) 1 for im ( x ) = 0 ∧ x mod π = 0 otherwise ) − ( ( { 0 c s c 2 ( x ) 1 for im ( x ) = 0 ∧ x mod π = 0 otherwise ) ( { 1 c s c 2 ( − x + 2 π ) 1 for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) ) + ⎩ ⎨ ⎧ 1 c s c 8 ( 2 x ) ( c s c 2 ( − 2 x + 2 π ) c s c 2 ( 2 x ) − 1 ) 4 for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise 4 4/ pi\ 6 2 2/ pi\
sin (x) + sin |x + --| - sin (x) - sin (x)*sin |x + --|
\ 2 / \ 2 / − sin 6 ( x ) + sin 4 ( x ) − sin 2 ( x ) sin 2 ( x + π 2 ) + sin 4 ( x + π 2 ) - \sin^{6}{\left(x \right)} + \sin^{4}{\left(x \right)} - \sin^{2}{\left(x \right)} \sin^{2}{\left(x + \frac{\pi}{2} \right)} + \sin^{4}{\left(x + \frac{\pi}{2} \right)} − sin 6 ( x ) + sin 4 ( x ) − sin 2 ( x ) sin 2 ( x + 2 π ) + sin 4 ( x + 2 π ) 6/x\ 6/pi x\ 8/pi x\ 4/x\ 8/x\ 6/x\ 6/pi x\
- 68*sin |-| - 8*sin |-- + -| + 9*sin |-- + -| + 30*sin |-| + 39*sin |-| - 64*sin |-|*sin |-- + -|
\2/ \2 2/ \2 2/ \2/ \2/ \2/ \2 2/ 39 sin 8 ( x 2 ) − 64 sin 6 ( x 2 ) sin 6 ( x 2 + π 2 ) − 68 sin 6 ( x 2 ) + 30 sin 4 ( x 2 ) + 9 sin 8 ( x 2 + π 2 ) − 8 sin 6 ( x 2 + π 2 ) 39 \sin^{8}{\left(\frac{x}{2} \right)} - 64 \sin^{6}{\left(\frac{x}{2} \right)} \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} - 68 \sin^{6}{\left(\frac{x}{2} \right)} + 30 \sin^{4}{\left(\frac{x}{2} \right)} + 9 \sin^{8}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} - 8 \sin^{6}{\left(\frac{x}{2} + \frac{\pi}{2} \right)} 39 sin 8 ( 2 x ) − 64 sin 6 ( 2 x ) sin 6 ( 2 x + 2 π ) − 68 sin 6 ( 2 x ) + 30 sin 4 ( 2 x ) + 9 sin 8 ( 2 x + 2 π ) − 8 sin 6 ( 2 x + 2 π ) 1 sec 6 ( x ) \frac{1}{\sec^{6}{\left(x \right)}} sec 6 ( x ) 1 / 1 for And(im(x) = 0, x mod 2*pi = 0)
|
| 6
{ 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( cot 2 ( x 2 ) − 1 ) 6 sin 12 ( x 2 ) otherwise \begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases} { 1 ( cot 2 ( 2 x ) − 1 ) 6 sin 12 ( 2 x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| |
// 0 for And(im(x) = 0, x mod pi = 0)\ // 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)\ || 4 |
|| | || | || | || | ||/ 2 \ |
- |< 6 | - |< 2 |*|< 2/ pi\ | + |< 4 | + |<| sin (x) | 8/x\ |
||sin (x) otherwise | ||sin (x) otherwise | ||sin |x + --| otherwise | ||sin (x) otherwise | |||-1 + ---------| *sin |-| otherwise |
\\ / \\ / \\ \ 2 / / \\ / ||| 4/x\| \2/ |
||| 4*sin |-|| |
\\\ \2// / ( − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 2 ( x ) otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 sin 2 ( x + π 2 ) otherwise ) ) + ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 4 ( x ) otherwise ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 6 ( x ) otherwise ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( − 1 + sin 2 ( x ) 4 sin 4 ( x 2 ) ) 4 sin 8 ( x 2 ) otherwise ) \left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\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^{2}{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{4}{\left(x \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\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 \\\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) ( − ( { 0 sin 2 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) ( { 1 sin 2 ( x + 2 π ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) ) + ( { 0 sin 4 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) − ( { 0 sin 6 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) + ⎩ ⎨ ⎧ 1 ( − 1 + 4 s i n 4 ( 2 x ) s i n 2 ( x ) ) 4 sin 8 ( 2 x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| |
// 0 for And(im(x) = 0, x mod pi = 0)\ // 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)\ || 4 |
|| | || | || | || | ||/ 2 \ |
- |< 6 | - |< 2 |*|< 2 | + |< 4 | + |<| sin (x) | 8/x\ |
||sin (x) otherwise | ||sin (x) otherwise | ||cos (x) otherwise | ||sin (x) otherwise | |||-1 + ---------| *sin |-| otherwise |
\\ / \\ / \\ / \\ / ||| 4/x\| \2/ |
||| 4*sin |-|| |
\\\ \2// / ( − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 2 ( x ) otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 cos 2 ( x ) otherwise ) ) + ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 4 ( x ) otherwise ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 6 ( x ) otherwise ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( − 1 + sin 2 ( x ) 4 sin 4 ( x 2 ) ) 4 sin 8 ( x 2 ) otherwise ) \left(- \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\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^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{4}{\left(x \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\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 \\\left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right)^{4} \sin^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) ( − ( { 0 sin 2 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) ( { 1 cos 2 ( x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) ) + ( { 0 sin 4 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) − ( { 0 sin 6 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) + ⎩ ⎨ ⎧ 1 ( − 1 + 4 s i n 4 ( 2 x ) s i n 2 ( x ) ) 4 sin 8 ( 2 x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise 1 1 1 1
------- + ------- - ------- - ---------------
4 4 6 2 2
csc (x) sec (x) csc (x) csc (x)*sec (x) 1 sec 4 ( x ) − 1 csc 2 ( x ) sec 2 ( x ) + 1 csc 4 ( x ) − 1 csc 6 ( x ) \frac{1}{\sec^{4}{\left(x \right)}} - \frac{1}{\csc^{2}{\left(x \right)} \sec^{2}{\left(x \right)}} + \frac{1}{\csc^{4}{\left(x \right)}} - \frac{1}{\csc^{6}{\left(x \right)}} sec 4 ( x ) 1 − csc 2 ( x ) sec 2 ( x ) 1 + csc 4 ( x ) 1 − csc 6 ( x ) 1 4 4/ pi\ 6/ pi\ 2 2/ pi\
cos (x) + cos |x - --| - cos |x - --| - cos (x)*cos |x - --|
\ 2 / \ 2 / \ 2 / cos 4 ( x ) − cos 2 ( x ) cos 2 ( x − π 2 ) − cos 6 ( x − π 2 ) + cos 4 ( x − π 2 ) \cos^{4}{\left(x \right)} - \cos^{2}{\left(x \right)} \cos^{2}{\left(x - \frac{\pi}{2} \right)} - \cos^{6}{\left(x - \frac{\pi}{2} \right)} + \cos^{4}{\left(x - \frac{\pi}{2} \right)} cos 4 ( x ) − cos 2 ( x ) cos 2 ( x − 2 π ) − cos 6 ( x − 2 π ) + cos 4 ( x − 2 π ) // / x \\ // / x \\ // / x \\ // / x \\ // / x \\ // / x \\ // / x \\
|| 0 for And|im(x) = 0, - mod pi = 0|| || 1 for And|im(x) = 0, - mod 2*pi = 0|| || 1 for And|im(x) = 0, - mod 2*pi = 0|| || 0 for And|im(x) = 0, - mod pi = 0|| || 0 for And|im(x) = 0, - mod pi = 0|| || 0 for And|im(x) = 0, - mod pi = 0|| || 1 for And|im(x) = 0, - mod 2*pi = 0||
|| \ 2 /| || \ 2 /| || \ 2 /| || \ 2 /| || \ 2 /| || \ 2 /| || \ 2 /|
|| | || | || | || | || | || | || |
|| 6/x\ | || 6 | || 8 | || 4/x\ | || 8/x\ | || 6/x\ | || 6 |
|| 64*cot |-| | ||/ 2/x\\ | ||/ 2/x\\ | || 16*cot |-| | || 256*cot |-| | || 64*cot |-| | ||/ 2/x\\ |
- 68*|< \4/ | - 8*|<|-1 + cot |-|| | + 9*|<|-1 + cot |-|| | + 30*|< \4/ | + 39*|< \4/ | - 64*|< \4/ |*|<|-1 + cot |-|| |
||-------------- otherwise | ||\ \4// | ||\ \4// | ||-------------- otherwise | ||-------------- otherwise | ||-------------- otherwise | ||\ \4// |
|| 6 | ||--------------- otherwise | ||--------------- otherwise | || 4 | || 8 | || 6 | ||--------------- otherwise |
||/ 2/x\\ | || 6 | || 8 | ||/ 2/x\\ | ||/ 2/x\\ | ||/ 2/x\\ | || 6 |
|||1 + cot |-|| | || / 2/x\\ | || / 2/x\\ | |||1 + cot |-|| | |||1 + cot |-|| | |||1 + cot |-|| | || / 2/x\\ |
||\ \4// | || |1 + cot |-|| | || |1 + cot |-|| | ||\ \4// | ||\ \4// | ||\ \4// | || |1 + cot |-|| |
\\ / \\ \ \4// / \\ \ \4// / \\ / \\ / \\ / \\ \ \4// / ( 39 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 256 cot 8 ( x 4 ) ( cot 2 ( x 4 ) + 1 ) 8 otherwise ) ) − ( 64 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 64 cot 6 ( x 4 ) ( cot 2 ( x 4 ) + 1 ) 6 otherwise ) ( { 1 for im ( x ) = 0 ∧ x 2 m o d 2 π = 0 ( cot 2 ( x 4 ) − 1 ) 6 ( cot 2 ( x 4 ) + 1 ) 6 otherwise ) ) − ( 68 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 64 cot 6 ( x 4 ) ( cot 2 ( x 4 ) + 1 ) 6 otherwise ) ) + ( 30 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 16 cot 4 ( x 4 ) ( cot 2 ( x 4 ) + 1 ) 4 otherwise ) ) − ( 8 ( { 1 for im ( x ) = 0 ∧ x 2 m o d 2 π = 0 ( cot 2 ( x 4 ) − 1 ) 6 ( cot 2 ( x 4 ) + 1 ) 6 otherwise ) ) + ( 9 ( { 1 for im ( x ) = 0 ∧ x 2 m o d 2 π = 0 ( cot 2 ( x 4 ) − 1 ) 8 ( cot 2 ( x 4 ) + 1 ) 8 otherwise ) ) \left(39 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)\right) - \left(64 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)\right) - \left(68 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)\right) + \left(30 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right)\right) + \left(9 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{8}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right)\right) 39 ⎩ ⎨ ⎧ 0 ( c o t 2 ( 4 x ) + 1 ) 8 256 c o t 8 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise − 64 ⎩ ⎨ ⎧ 0 ( c o t 2 ( 4 x ) + 1 ) 6 64 c o t 6 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise ⎩ ⎨ ⎧ 1 ( c o t 2 ( 4 x ) + 1 ) 6 ( c o t 2 ( 4 x ) − 1 ) 6 for im ( x ) = 0 ∧ 2 x mod 2 π = 0 otherwise − 68 ⎩ ⎨ ⎧ 0 ( c o t 2 ( 4 x ) + 1 ) 6 64 c o t 6 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise + 30 ⎩ ⎨ ⎧ 0 ( c o t 2 ( 4 x ) + 1 ) 4 16 c o t 4 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise − 8 ⎩ ⎨ ⎧ 1 ( c o t 2 ( 4 x ) + 1 ) 6 ( c o t 2 ( 4 x ) − 1 ) 6 for im ( x ) = 0 ∧ 2 x mod 2 π = 0 otherwise + 9 ⎩ ⎨ ⎧ 1 ( c o t 2 ( 4 x ) + 1 ) 8 ( c o t 2 ( 4 x ) − 1 ) 8 for im ( x ) = 0 ∧ 2 x mod 2 π = 0 otherwise // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| |
|| 4 |
||/ 2/x pi\\ |
// 0 for And(im(x) = 0, x mod pi = 0)\ // 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)\ ||| sec |- - --|| |
|| | || | || | || | ||| \2 2 /| |
|| 1 | || 1 | || 1 | || 16 | |||-1 + ------------| |
- |<------------ otherwise | - |<------------ otherwise |*|<------- otherwise | + |<-------------------- otherwise | + |<| 2/x\ | |
|| 6/ pi\ | || 2/ pi\ | || 2 | || 4/x\ 4/x pi\ | ||| sec |-| | |
||sec |x - --| | ||sec |x - --| | ||sec (x) | ||sec |-|*sec |- - --| | ||\ \2/ / |
\\ \ 2 / / \\ \ 2 / / \\ / \\ \2/ \2 2 / / ||-------------------- otherwise |
|| 8/x pi\ |
|| sec |- - --| |
|| \2 2 / |
\\ / ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 16 sec 4 ( x 2 ) sec 4 ( x 2 − π 2 ) otherwise ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 1 sec 6 ( x − π 2 ) otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 1 sec 2 ( x − π 2 ) otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 1 sec 2 ( x ) otherwise ) ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( − 1 + sec 2 ( x 2 − π 2 ) sec 2 ( x 2 ) ) 4 sec 8 ( x 2 − π 2 ) otherwise ) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{16}{\sec^{4}{\left(\frac{x}{2} \right)} \sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec^{6}{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec^{2}{\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^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)^{4}}{\sec^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) ( { 0 s e c 4 ( 2 x ) s e c 4 ( 2 x − 2 π ) 16 for im ( x ) = 0 ∧ x mod π = 0 otherwise ) − ( { 0 s e c 6 ( x − 2 π ) 1 for im ( x ) = 0 ∧ x mod π = 0 otherwise ) − ( ( { 0 s e c 2 ( x − 2 π ) 1 for im ( x ) = 0 ∧ x mod π = 0 otherwise ) ( { 1 s e c 2 ( x ) 1 for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) ) + ⎩ ⎨ ⎧ 1 s e c 8 ( 2 x − 2 π ) ( − 1 + s e c 2 ( 2 x ) s e c 2 ( 2 x − 2 π ) ) 4 for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise 6 8 6
12/x\ 6/x\ / 2/x\\ 12/x\ / 2/x\\ 16/x\ 8/x\ 4/x\ 16/x\ 8/x\ / 2/x\\ 24/x\ 6/x\
- 4352*cos |-|*tan |-| - 8*|1 - tan |-|| *cos |-| + 9*|1 - tan |-|| *cos |-| + 480*cos |-|*tan |-| + 9984*cos |-|*tan |-| - 4096*|1 - tan |-|| *cos |-|*tan |-|
\4/ \4/ \ \4// \4/ \ \4// \4/ \4/ \4/ \4/ \4/ \ \4// \4/ \4/ 9 ( 1 − tan 2 ( x 4 ) ) 8 cos 16 ( x 4 ) − 4096 ( 1 − tan 2 ( x 4 ) ) 6 cos 24 ( x 4 ) tan 6 ( x 4 ) − 8 ( 1 − tan 2 ( x 4 ) ) 6 cos 12 ( x 4 ) + 9984 cos 16 ( x 4 ) tan 8 ( x 4 ) − 4352 cos 12 ( x 4 ) tan 6 ( x 4 ) + 480 cos 8 ( x 4 ) tan 4 ( x 4 ) 9 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{8} \cos^{16}{\left(\frac{x}{4} \right)} - 4096 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \cos^{24}{\left(\frac{x}{4} \right)} \tan^{6}{\left(\frac{x}{4} \right)} - 8 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \cos^{12}{\left(\frac{x}{4} \right)} + 9984 \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)} - 4352 \cos^{12}{\left(\frac{x}{4} \right)} \tan^{6}{\left(\frac{x}{4} \right)} + 480 \cos^{8}{\left(\frac{x}{4} \right)} \tan^{4}{\left(\frac{x}{4} \right)} 9 ( 1 − tan 2 ( 4 x ) ) 8 cos 16 ( 4 x ) − 4096 ( 1 − tan 2 ( 4 x ) ) 6 cos 24 ( 4 x ) tan 6 ( 4 x ) − 8 ( 1 − tan 2 ( 4 x ) ) 6 cos 12 ( 4 x ) + 9984 cos 16 ( 4 x ) tan 8 ( 4 x ) − 4352 cos 12 ( 4 x ) tan 6 ( 4 x ) + 480 cos 8 ( 4 x ) tan 4 ( 4 x ) 6/x pi\ 6/x\ 8/x\ 4/x pi\ 8/x pi\ 6/x\ 6/x pi\
- 68*cos |- - --| - 8*cos |-| + 9*cos |-| + 30*cos |- - --| + 39*cos |- - --| - 64*cos |-|*cos |- - --|
\2 2 / \2/ \2/ \2 2 / \2 2 / \2/ \2 2 / 9 cos 8 ( x 2 ) − 64 cos 6 ( x 2 ) cos 6 ( x 2 − π 2 ) − 8 cos 6 ( x 2 ) + 39 cos 8 ( x 2 − π 2 ) − 68 cos 6 ( x 2 − π 2 ) + 30 cos 4 ( x 2 − π 2 ) 9 \cos^{8}{\left(\frac{x}{2} \right)} - 64 \cos^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} - 8 \cos^{6}{\left(\frac{x}{2} \right)} + 39 \cos^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} - 68 \cos^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} + 30 \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} 9 cos 8 ( 2 x ) − 64 cos 6 ( 2 x ) cos 6 ( 2 x − 2 π ) − 8 cos 6 ( 2 x ) + 39 cos 8 ( 2 x − 2 π ) − 68 cos 6 ( 2 x − 2 π ) + 30 cos 4 ( 2 x − 2 π ) // 1 for And(im(x) = 0, x mod 2*pi = 0)\
// 0 for And(im(x) = 0, x mod pi = 0)\ // 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)\ || |
|| | || | || | || | || 4 |
|| 6/x\ | || 2/x\ | || 2 | || 8/x\ | || / 1 \ 8/x\ |
|| 64*tan |-| | || 4*tan |-| | ||/ 2/x\\ | || 4096*tan |-| | ||256*|-1 + -------| *tan |-| |
|| \2/ | || \2/ | |||1 - tan |-|| | || \4/ | || | 2/x\| \4/ |
- |<-------------- otherwise | - |<-------------- otherwise |*|<\ \2// | + |<---------------------- otherwise | + |< | tan |-|| |
|| 6 | || 2 | ||-------------- otherwise | || 8 | || \ \2// |
||/ 2/x\\ | ||/ 2/x\\ | || 2 | ||/ 2/x\\ 4/x\ | ||--------------------------- otherwise |
|||1 + tan |-|| | |||1 + tan |-|| | ||/ 2/x\\ | |||1 + tan |-|| *tan |-| | || 8 |
||\ \2// | ||\ \2// | |||1 + tan |-|| | ||\ \4// \2/ | || / 2/x\\ |
\\ / \\ / \\\ \2// / \\ / || |1 + tan |-|| |
\\ \ \4// / ( − { 0 for im ( x ) = 0 ∧ x m o d π = 0 64 tan 6 ( x 2 ) ( tan 2 ( x 2 ) + 1 ) 6 otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 4 tan 2 ( x 2 ) ( tan 2 ( x 2 ) + 1 ) 2 otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( 1 − tan 2 ( x 2 ) ) 2 ( tan 2 ( x 2 ) + 1 ) 2 otherwise ) ) + ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 4096 tan 8 ( x 4 ) ( tan 2 ( x 4 ) + 1 ) 8 tan 4 ( x 2 ) otherwise ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 256 ( − 1 + 1 tan 2 ( x 2 ) ) 4 tan 8 ( x 4 ) ( tan 2 ( x 4 ) + 1 ) 8 otherwise ) \left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4096 \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8} \tan^{4}{\left(\frac{x}{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{256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) − ⎩ ⎨ ⎧ 0 ( t a n 2 ( 2 x ) + 1 ) 6 64 t a n 6 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise − ⎩ ⎨ ⎧ 0 ( t a n 2 ( 2 x ) + 1 ) 2 4 t a n 2 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ⎩ ⎨ ⎧ 1 ( t a n 2 ( 2 x ) + 1 ) 2 ( 1 − t a n 2 ( 2 x ) ) 2 for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise + ⎩ ⎨ ⎧ 0 ( t a n 2 ( 4 x ) + 1 ) 8 t a n 4 ( 2 x ) 4096 t a n 8 ( 4 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise + ⎩ ⎨ ⎧ 1 ( t a n 2 ( 4 x ) + 1 ) 8 256 ( − 1 + t a n 2 ( 2 x ) 1 ) 4 t a n 8 ( 4 x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise 1 1 1 1
------- + ------------ - ------- - --------------------
4 4/pi \ 6 2 2/pi \
csc (x) csc |-- - x| csc (x) csc (x)*csc |-- - x|
\2 / \2 / 1 csc 4 ( − x + π 2 ) − 1 csc 2 ( x ) csc 2 ( − x + π 2 ) + 1 csc 4 ( x ) − 1 csc 6 ( x ) \frac{1}{\csc^{4}{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc^{2}{\left(x \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{4}{\left(x \right)}} - \frac{1}{\csc^{6}{\left(x \right)}} csc 4 ( − x + 2 π ) 1 − csc 2 ( x ) csc 2 ( − x + 2 π ) 1 + csc 4 ( x ) 1 − csc 6 ( x ) 1 6
/ 2/x\\ 12/x\
|1 - tan |-|| *cos |-|
\ \2// \2/ ( 1 − tan 2 ( x 2 ) ) 6 cos 12 ( x 2 ) \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{6} \cos^{12}{\left(\frac{x}{2} \right)} ( 1 − tan 2 ( 2 x ) ) 6 cos 12 ( 2 x ) // / x \\ // / x \\ // / x \\
// / x \\ || 1 for And|im(x) = 0, - mod 2*pi = 0|| || 1 for And|im(x) = 0, - mod 2*pi = 0|| // / x \\ // / x \\ // / x \\ || 1 for And|im(x) = 0, - mod 2*pi = 0||
|| 0 for And|im(x) = 0, - mod pi = 0|| || \ 2 /| || \ 2 /| || 0 for And|im(x) = 0, - mod pi = 0|| || 0 for And|im(x) = 0, - mod pi = 0|| || 0 for And|im(x) = 0, - mod pi = 0|| || \ 2 /|
|| \ 2 /| || | || | || \ 2 /| || \ 2 /| || \ 2 /| || |
- 68*|< | - 8*|< 6 | + 9*|< 8 | + 30*|< | + 39*|< | - 64*|< |*|< 6 |
|| 6/x\ 12/x\ | ||/ 2/x\\ 12/x\ | ||/ 2/x\\ 16/x\ | || 4/x\ 8/x\ | || 8/x\ 16/x\ | || 6/x\ 12/x\ | ||/ 2/x\\ 12/x\ |
||64*cot |-|*sin |-| otherwise | |||-1 + cot |-|| *sin |-| otherwise | |||-1 + cot |-|| *sin |-| otherwise | ||16*cot |-|*sin |-| otherwise | ||256*cot |-|*sin |-| otherwise | ||64*cot |-|*sin |-| otherwise | |||-1 + cot |-|| *sin |-| otherwise |
\\ \4/ \4/ / ||\ \4// \4/ | ||\ \4// \4/ | \\ \4/ \4/ / \\ \4/ \4/ / \\ \4/ \4/ / ||\ \4// \4/ |
\\ / \\ / \\ / ( 30 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 16 sin 8 ( x 4 ) cot 4 ( x 4 ) otherwise ) ) − ( 64 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 64 sin 12 ( x 4 ) cot 6 ( x 4 ) otherwise ) ( { 1 for im ( x ) = 0 ∧ x 2 m o d 2 π = 0 ( cot 2 ( x 4 ) − 1 ) 6 sin 12 ( x 4 ) otherwise ) ) − ( 68 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 64 sin 12 ( x 4 ) cot 6 ( x 4 ) otherwise ) ) + ( 39 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 256 sin 16 ( x 4 ) cot 8 ( x 4 ) otherwise ) ) − ( 8 ( { 1 for im ( x ) = 0 ∧ x 2 m o d 2 π = 0 ( cot 2 ( x 4 ) − 1 ) 6 sin 12 ( x 4 ) otherwise ) ) + ( 9 ( { 1 for im ( x ) = 0 ∧ x 2 m o d 2 π = 0 ( cot 2 ( x 4 ) − 1 ) 8 sin 16 ( x 4 ) otherwise ) ) \left(30 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\16 \sin^{8}{\left(\frac{x}{4} \right)} \cot^{4}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(64 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\64 \sin^{12}{\left(\frac{x}{4} \right)} \cot^{6}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(68 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\64 \sin^{12}{\left(\frac{x}{4} \right)} \cot^{6}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(39 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(8 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{6} \sin^{12}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(9 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{8} \sin^{16}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right)\right) ( 30 ( { 0 16 sin 8 ( 4 x ) cot 4 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise ) ) − ( 64 ( { 0 64 sin 12 ( 4 x ) cot 6 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise ) ( { 1 ( cot 2 ( 4 x ) − 1 ) 6 sin 12 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod 2 π = 0 otherwise ) ) − ( 68 ( { 0 64 sin 12 ( 4 x ) cot 6 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise ) ) + ( 39 ( { 0 256 sin 16 ( 4 x ) cot 8 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise ) ) − ( 8 ( { 1 ( cot 2 ( 4 x ) − 1 ) 6 sin 12 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod 2 π = 0 otherwise ) ) + ( 9 ( { 1 ( cot 2 ( 4 x ) − 1 ) 8 sin 16 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod 2 π = 0 otherwise ) ) cos 6 ( x ) \cos^{6}{\left(x \right)} cos 6 ( x ) 68 8 9 30 39 64
- ------------ - ------- + ------- + ------------ + ------------ - --------------------
6/x pi\ 6/x\ 8/x\ 4/x pi\ 8/x pi\ 6/x\ 6/x pi\
sec |- - --| sec |-| sec |-| sec |- - --| sec |- - --| sec |-|*sec |- - --|
\2 2 / \2/ \2/ \2 2 / \2 2 / \2/ \2 2 / 30 sec 4 ( x 2 − π 2 ) − 68 sec 6 ( x 2 − π 2 ) + 39 sec 8 ( x 2 − π 2 ) − 8 sec 6 ( x 2 ) − 64 sec 6 ( x 2 ) sec 6 ( x 2 − π 2 ) + 9 sec 8 ( x 2 ) \frac{30}{\sec^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{68}{\sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{39}{\sec^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{8}{\sec^{6}{\left(\frac{x}{2} \right)}} - \frac{64}{\sec^{6}{\left(\frac{x}{2} \right)} \sec^{6}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + \frac{9}{\sec^{8}{\left(\frac{x}{2} \right)}} sec 4 ( 2 x − 2 π ) 30 − sec 6 ( 2 x − 2 π ) 68 + sec 8 ( 2 x − 2 π ) 39 − sec 6 ( 2 x ) 8 − sec 6 ( 2 x ) sec 6 ( 2 x − 2 π ) 64 + sec 8 ( 2 x ) 9 // 1 for And(im(x) = 0, x mod 2*pi = 0)\
// 0 for And(im(x) = 0, x mod pi = 0)\ // 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)\ || |
|| | || | || | || | || 4 |
- |< 6 | - |< 2 |*|< 2 | + |< 4/x\ 4/x\ | + | ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 16 sin 4 ( x 2 ) cos 4 ( x 2 ) otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 2 ( x ) otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 cos 2 ( x ) otherwise ) ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 6 ( x ) otherwise ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( 1 − tan 2 ( x 2 ) ) 4 cos 8 ( x 2 ) otherwise ) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \sin^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\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^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\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 \\\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) ( { 0 16 sin 4 ( 2 x ) cos 4 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) − ( ( { 0 sin 2 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) ( { 1 cos 2 ( x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) ) − ( { 0 sin 6 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) + ( { 1 ( 1 − tan 2 ( 2 x ) ) 4 cos 8 ( 2 x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) 6/x\ 6/x\ 8/x\ 4/x\ 8/x\ 6/x\ 6/x\
- 68*sin |-| - 8*cos |-| + 9*cos |-| + 30*sin |-| + 39*sin |-| - 64*cos |-|*sin |-|
\2/ \2/ \2/ \2/ \2/ \2/ \2/ 39 sin 8 ( x 2 ) − 64 sin 6 ( x 2 ) cos 6 ( x 2 ) − 68 sin 6 ( x 2 ) + 30 sin 4 ( x 2 ) + 9 cos 8 ( x 2 ) − 8 cos 6 ( x 2 ) 39 \sin^{8}{\left(\frac{x}{2} \right)} - 64 \sin^{6}{\left(\frac{x}{2} \right)} \cos^{6}{\left(\frac{x}{2} \right)} - 68 \sin^{6}{\left(\frac{x}{2} \right)} + 30 \sin^{4}{\left(\frac{x}{2} \right)} + 9 \cos^{8}{\left(\frac{x}{2} \right)} - 8 \cos^{6}{\left(\frac{x}{2} \right)} 39 sin 8 ( 2 x ) − 64 sin 6 ( 2 x ) cos 6 ( 2 x ) − 68 sin 6 ( 2 x ) + 30 sin 4 ( 2 x ) + 9 cos 8 ( 2 x ) − 8 cos 6 ( 2 x ) // 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)\ // 0 for And(im(x) = 0, x mod pi = 0)\ // 1 for And(im(x) = 0, x mod 2*pi = 0)\ || 16/x\ 8/x\ | || 4 |
|| | || | || | ||4096*cos |-|*tan |-| | || / 1 \ 16/x\ 8/x\ |
- |< 6 | - |< 2 |*|< 2 | + |< \4/ \4/ | + |<256*|-1 + -------| *cos |-|*tan |-| otherwise |
||sin (x) otherwise | ||sin (x) otherwise | ||cos (x) otherwise | ||--------------------- otherwise | || | 2/x\| \4/ \4/ |
\\ / \\ / \\ / || 4/x\ | || | tan |-|| |
|| tan |-| | || \ \2// |
\\ \2/ / \\ / ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 4096 cos 16 ( x 4 ) tan 8 ( x 4 ) tan 4 ( x 2 ) otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 2 ( x ) otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 cos 2 ( x ) otherwise ) ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 6 ( x ) otherwise ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 256 ( − 1 + 1 tan 2 ( x 2 ) ) 4 cos 16 ( x 4 ) tan 8 ( x 4 ) otherwise ) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4096 \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)}}{\tan^{4}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{2}{\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^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin^{6}{\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 \\256 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right)^{4} \cos^{16}{\left(\frac{x}{4} \right)} \tan^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) ⎩ ⎨ ⎧ 0 t a n 4 ( 2 x ) 4096 c o s 16 ( 4 x ) t a n 8 ( 4 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise − ( ( { 0 sin 2 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) ( { 1 cos 2 ( x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) ) − ( { 0 sin 6 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) + ⎩ ⎨ ⎧ 1 256 ( − 1 + t a n 2 ( 2 x ) 1 ) 4 cos 16 ( 4 x ) tan 8 ( 4 x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise 68 8 9 30 39 64
- ------- - ------------ + ------------ + ------- + ------- - --------------------
6/x\ 6/pi x\ 8/pi x\ 4/x\ 8/x\ 6/x\ 6/pi x\
csc |-| csc |-- - -| csc |-- - -| csc |-| csc |-| csc |-|*csc |-- - -|
\2/ \2 2/ \2 2/ \2/ \2/ \2/ \2 2/ − 8 csc 6 ( − x 2 + π 2 ) + 9 csc 8 ( − x 2 + π 2 ) + 30 csc 4 ( x 2 ) − 68 csc 6 ( x 2 ) − 64 csc 6 ( x 2 ) csc 6 ( − x 2 + π 2 ) + 39 csc 8 ( x 2 ) - \frac{8}{\csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{9}{\csc^{8}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{30}{\csc^{4}{\left(\frac{x}{2} \right)}} - \frac{68}{\csc^{6}{\left(\frac{x}{2} \right)}} - \frac{64}{\csc^{6}{\left(\frac{x}{2} \right)} \csc^{6}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{39}{\csc^{8}{\left(\frac{x}{2} \right)}} − csc 6 ( − 2 x + 2 π ) 8 + csc 8 ( − 2 x + 2 π ) 9 + csc 4 ( 2 x ) 30 − csc 6 ( 2 x ) 68 − csc 6 ( 2 x ) csc 6 ( − 2 x + 2 π ) 64 + csc 8 ( 2 x ) 39 // 1 for And(im(x) = 0, x mod 2*pi = 0)\
|| |
|| 4 |
// 0 for And(im(x) = 0, x mod pi = 0)\ // 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)\ ||/ 2/x\ \ |
|| | || | || | || | ||| cos |-| | |
- |< 6/ pi\ | - |< 2/ pi\ |*|< 2 | + |< 4/x\ 4/x pi\ | + |<| \2/ | 8/x pi\ |
||cos |x - --| otherwise | ||cos |x - --| otherwise | ||cos (x) otherwise | ||16*cos |-|*cos |- - --| otherwise | |||-1 + ------------| *cos |- - --| otherwise |
\\ \ 2 / / \\ \ 2 / / \\ / \\ \2/ \2 2 / / ||| 2/x pi\| \2 2 / |
||| cos |- - --|| |
||\ \2 2 // |
\\ / ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 16 cos 4 ( x 2 ) cos 4 ( x 2 − π 2 ) otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 cos 2 ( x − π 2 ) otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 cos 2 ( x ) otherwise ) ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 cos 6 ( x − π 2 ) otherwise ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( cos 2 ( x 2 ) cos 2 ( x 2 − π 2 ) − 1 ) 4 cos 8 ( x 2 − π 2 ) otherwise ) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \cos^{4}{\left(\frac{x}{2} \right)} \cos^{4}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos^{2}{\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^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos^{6}{\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 \\\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{4} \cos^{8}{\left(\frac{x}{2} - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) ( { 0 16 cos 4 ( 2 x ) cos 4 ( 2 x − 2 π ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) − ( ( { 0 cos 2 ( x − 2 π ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) ( { 1 cos 2 ( x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise ) ) − ( { 0 cos 6 ( x − 2 π ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ) + ⎩ ⎨ ⎧ 1 ( c o s 2 ( 2 x − 2 π ) c o s 2 ( 2 x ) − 1 ) 4 cos 8 ( 2 x − 2 π ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise // 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)\ // 0 for And(im(x) = 0, x mod pi = 0)\ // 1 for And(im(x) = 0, x mod 2*pi = 0)\ || | || |
|| | || | || | || // / x \\ | || // / x \\ |
||/ 0 for And(im(x) = 0, x mod pi = 0) | ||/ 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, - mod pi = 0|| | || || 0 for And|im(x) = 0, - mod pi = 0|| |
||| | ||| | ||| | || || \ 2 /| | || || \ 2 /| |
||| 6/x\ | ||| 2/x\ | ||| 2 | || || | | || || | |
||| 64*cot |-| | ||| 4*cot |-| | |||/ 2/x\\ | || || 8/x\ | | || 4 || 8/x\ | |
- |<| \2/ | - |<| \2/ |*|<||-1 + cot |-|| | + |< 4/x\ || 256*cot |-| | | + | ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 16 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 256 cot 8 ( x 4 ) ( cot 2 ( x 4 ) + 1 ) 8 otherwise ) cot 4 ( x 2 ) otherwise ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 { 0 for im ( x ) = 0 ∧ x m o d π = 0 64 cot 6 ( x 2 ) ( cot 2 ( x 2 ) + 1 ) 6 otherwise otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 { 0 for im ( x ) = 0 ∧ x m o d π = 0 4 cot 2 ( x 2 ) ( cot 2 ( x 2 ) + 1 ) 2 otherwise otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( cot 2 ( x 2 ) − 1 ) 2 ( cot 2 ( x 2 ) + 1 ) 2 otherwise otherwise ) ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( cot 2 ( x 2 ) − 1 ) 4 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 256 cot 8 ( x 4 ) ( cot 2 ( x 4 ) + 1 ) 8 otherwise ) otherwise ) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \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{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\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{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \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{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\\frac{256 \cot^{8}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) ⎩ ⎨ ⎧ 0 16 ⎩ ⎨ ⎧ 0 ( c o t 2 ( 4 x ) + 1 ) 8 256 c o t 8 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise cot 4 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise − ⎩ ⎨ ⎧ 0 ⎩ ⎨ ⎧ 0 ( c o t 2 ( 2 x ) + 1 ) 6 64 c o t 6 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise for im ( x ) = 0 ∧ x mod π = 0 otherwise − ⎩ ⎨ ⎧ 0 ⎩ ⎨ ⎧ 0 ( c o t 2 ( 2 x ) + 1 ) 2 4 c o t 2 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise for im ( x ) = 0 ∧ x mod π = 0 otherwise ⎩ ⎨ ⎧ 1 ⎩ ⎨ ⎧ 1 ( c o t 2 ( 2 x ) + 1 ) 2 ( c o t 2 ( 2 x ) − 1 ) 2 for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise + ⎩ ⎨ ⎧ 1 ( cot 2 ( 2 x ) − 1 ) 4 ⎩ ⎨ ⎧ 0 ( c o t 2 ( 4 x ) + 1 ) 8 256 c o t 8 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise // 0 for And(im(x) = 0, x mod pi = 0)\ // 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)\
|| | || | || | || | || |
|| 6/x\ | || 2/x\ | || 2 | || 4/x\ | || 4 |
|| 64*cot |-| | || 4*cot |-| | ||/ 2/x\\ | || 16*cot |-| | ||/ 2/x\\ |
|| \2/ | || \2/ | |||-1 + cot |-|| | || \2/ | |||-1 + cot |-|| |
- |<-------------- otherwise | - |<-------------- otherwise |*|<\ \2// | + |<-------------- otherwise | + |<\ \2// |
|| 6 | || 2 | ||--------------- otherwise | || 4 | ||--------------- otherwise |
||/ 2/x\\ | ||/ 2/x\\ | || 2 | ||/ 2/x\\ | || 4 |
|||1 + cot |-|| | |||1 + cot |-|| | || / 2/x\\ | |||1 + cot |-|| | || / 2/x\\ |
||\ \2// | ||\ \2// | || |1 + cot |-|| | ||\ \2// | || |1 + cot |-|| |
\\ / \\ / \\ \ \2// / \\ / \\ \ \2// / ( − { 0 for im ( x ) = 0 ∧ x m o d π = 0 64 cot 6 ( x 2 ) ( cot 2 ( x 2 ) + 1 ) 6 otherwise ) + ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 16 cot 4 ( x 2 ) ( cot 2 ( x 2 ) + 1 ) 4 otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 4 cot 2 ( x 2 ) ( cot 2 ( x 2 ) + 1 ) 2 otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( cot 2 ( x 2 ) − 1 ) 2 ( cot 2 ( x 2 ) + 1 ) 2 otherwise ) ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( cot 2 ( x 2 ) − 1 ) 4 ( cot 2 ( x 2 ) + 1 ) 4 otherwise ) \left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{64 \cot^{6}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{16 \cot^{4}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) - \left(\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} & \text{otherwise} \end{cases}\right) − ⎩ ⎨ ⎧ 0 ( c o t 2 ( 2 x ) + 1 ) 6 64 c o t 6 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise + ⎩ ⎨ ⎧ 0 ( c o t 2 ( 2 x ) + 1 ) 4 16 c o t 4 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise − ⎩ ⎨ ⎧ 0 ( c o t 2 ( 2 x ) + 1 ) 2 4 c o t 2 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise ⎩ ⎨ ⎧ 1 ( c o t 2 ( 2 x ) + 1 ) 2 ( c o t 2 ( 2 x ) − 1 ) 2 for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise + ⎩ ⎨ ⎧ 1 ( c o t 2 ( 2 x ) + 1 ) 4 ( c o t 2 ( 2 x ) − 1 ) 4 for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise / 1 for And(im(x) = 0, x mod 2*pi = 0)
|
| 6
|/ 2/x\\
||-1 + cot |-||
<\ \2//
|--------------- otherwise
| 6
| / 2/x\\
| |1 + cot |-||
\ \ \2// { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( cot 2 ( x 2 ) − 1 ) 6 ( cot 2 ( x 2 ) + 1 ) 6 otherwise \begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{6}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} & \text{otherwise} \end{cases} ⎩ ⎨ ⎧ 1 ( c o t 2 ( 2 x ) + 1 ) 6 ( c o t 2 ( 2 x ) − 1 ) 6 for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise 4 2
/ 2/x\\ 6/x\ 4/x\ / 2/x\\ 2/x\
|1 - tan |-|| 64*tan |-| 16*tan |-| 4*|1 - tan |-|| *tan |-|
\ \2// \2/ \2/ \ \2// \2/
-------------- - -------------- + -------------- - ------------------------
4 6 4 4
/ 2/x\\ / 2/x\\ / 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |-|| |1 + tan |-||
\ \2// \ \2// \ \2// \ \2// ( 1 − tan 2 ( x 2 ) ) 4 ( tan 2 ( x 2 ) + 1 ) 4 − 4 ( 1 − tan 2 ( x 2 ) ) 2 tan 2 ( x 2 ) ( tan 2 ( x 2 ) + 1 ) 4 + 16 tan 4 ( x 2 ) ( tan 2 ( x 2 ) + 1 ) 4 − 64 tan 6 ( x 2 ) ( tan 2 ( x 2 ) + 1 ) 6 \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{4}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} - \frac{4 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2} \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} + \frac{16 \tan^{4}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{4}} - \frac{64 \tan^{6}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{6}} ( tan 2 ( 2 x ) + 1 ) 4 ( 1 − tan 2 ( 2 x ) ) 4 − ( tan 2 ( 2 x ) + 1 ) 4 4 ( 1 − tan 2 ( 2 x ) ) 2 tan 2 ( 2 x ) + ( tan 2 ( 2 x ) + 1 ) 4 16 tan 4 ( 2 x ) − ( tan 2 ( 2 x ) + 1 ) 6 64 tan 6 ( 2 x ) // 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)\ // 0 for And(im(x) = 0, x mod pi = 0)\ // 1 for And(im(x) = 0, x mod 2*pi = 0)\ || | || |
|| | || | || | || // / x \\ | || // / x \\ |
||/ 0 for And(im(x) = 0, x mod pi = 0) | ||/ 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, - mod pi = 0|| | || 4 || 0 for And|im(x) = 0, - mod pi = 0|| |
- |<| | - |<| |*|<| | + |< 4/x\ || \ 2 /| | + | ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 16 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 256 sin 16 ( x 4 ) cot 8 ( x 4 ) otherwise ) cot 4 ( x 2 ) otherwise ) − ( ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 2 ( x ) otherwise otherwise ) ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 cos 2 ( x ) otherwise otherwise ) ) − ( { 0 for im ( x ) = 0 ∧ x m o d π = 0 { 0 for im ( x ) = 0 ∧ x m o d π = 0 sin 6 ( x ) otherwise otherwise ) + ( { 1 for im ( x ) = 0 ∧ x m o d 2 π = 0 ( cot 2 ( x 2 ) − 1 ) 4 ( { 0 for im ( x ) = 0 ∧ x 2 m o d π = 0 256 sin 16 ( x 4 ) cot 8 ( x 4 ) otherwise ) otherwise ) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\16 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) \cot^{4}{\left(\frac{x}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\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^{2}{\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^{2}{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \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^{6}{\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 \\\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{4} \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge \frac{x}{2} \bmod \pi = 0 \\256 \sin^{16}{\left(\frac{x}{4} \right)} \cot^{8}{\left(\frac{x}{4} \right)} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) ⎩ ⎨ ⎧ 0 16 ( { 0 256 sin 16 ( 4 x ) cot 8 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise ) cot 4 ( 2 x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise − ⎩ ⎨ ⎧ 0 { 0 sin 2 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise for im ( x ) = 0 ∧ x mod π = 0 otherwise ⎩ ⎨ ⎧ 1 { 1 cos 2 ( x ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise − ⎩ ⎨ ⎧ 0 { 0 sin 6 ( x ) for im ( x ) = 0 ∧ x mod π = 0 otherwise for im ( x ) = 0 ∧ x mod π = 0 otherwise + ⎩ ⎨ ⎧ 1 ( cot 2 ( 2 x ) − 1 ) 4 ( { 0 256 sin 16 ( 4 x ) cot 8 ( 4 x ) for im ( x ) = 0 ∧ 2 x mod π = 0 otherwise ) for im ( x ) = 0 ∧ x mod 2 π = 0 otherwise 6 8 6
6/x\ / 2/x\\ / 2/x\\ 4/x\ 8/x\ / 2/x\\ 6/x\
4352*tan |-| 8*|1 - tan |-|| 9*|1 - tan |-|| 480*tan |-| 9984*tan |-| 4096*|1 - tan |-|| *tan |-|
\4/ \ \4// \ \4// \4/ \4/ \ \4// \4/
- -------------- - ---------------- + ---------------- + -------------- + -------------- - ---------------------------
6 6 8 4 8 12
/ 2/x\\ / 2/x\\ / 2/x\\ / 2/x\\ / 2/x\\ / 2/x\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |-|| |1 + tan |-|| |1 + tan |-|| |1 + tan |-||
\ \4// \ \4// \ \4// \ \4// \ \4// \ \4// 9 ( 1 − tan 2 ( x 4 ) ) 8 ( tan 2 ( x 4 ) + 1 ) 8 − 8 ( 1 − tan 2 ( x 4 ) ) 6 ( tan 2 ( x 4 ) + 1 ) 6 − 4096 ( 1 − tan 2 ( x 4 ) ) 6 tan 6 ( x 4 ) ( tan 2 ( x 4 ) + 1 ) 12 + 480 tan 4 ( x 4 ) ( tan 2 ( x 4 ) + 1 ) 4 − 4352 tan 6 ( x 4 ) ( tan 2 ( x 4 ) + 1 ) 6 + 9984 tan 8 ( x 4 ) ( tan 2 ( x 4 ) + 1 ) 8 \frac{9 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{8}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} - \frac{8 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} - \frac{4096 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{6} \tan^{6}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{12}} + \frac{480 \tan^{4}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}} - \frac{4352 \tan^{6}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{6}} + \frac{9984 \tan^{8}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{8}} ( tan 2 ( 4 x ) + 1 ) 8 9 ( 1 − tan 2 ( 4 x ) ) 8 − ( tan 2 ( 4 x ) + 1 ) 6 8 ( 1 − tan 2 ( 4 x ) ) 6 − ( tan 2 ( 4 x ) + 1 ) 12 4096 ( 1 − tan 2 ( 4 x ) ) 6 tan 6 ( 4 x ) + ( tan 2 ( 4 x ) + 1 ) 4 480 tan 4 ( 4 x ) − ( tan 2 ( 4 x ) + 1 ) 6 4352 tan 6 ( 4 x ) + ( tan 2 ( 4 x ) + 1 ) 8 9984 tan 8 ( 4 x ) 6/ pi\
sin |x + --|
\ 2 / sin 6 ( x + π 2 ) \sin^{6}{\left(x + \frac{\pi}{2} \right)} sin 6 ( x + 2 π )