В настоящей статье приведены формулы дифференцирования и формулы разложения в степенные ряды для обратных гиперболических и обратных тригонометрических функций.
Для производной всюду используется сокращенное обозначение:d z \mathrm{d}_z d z d d z \frac{d}{dz} d z d
d z a r c t a n h z = ( 1 − z 2 ) − 1 , d z arctan z = ( 1 + z 2 ) − 1 , \mathrm{d}_z \,\mathrm{arctanh}\, z = (1 -z^2)^{-1} \;,
\qquad
\mathrm{d}_z \,\arctan z = (1 +z^2)^{-1} \;,
d z a r c t a n h z = ( 1 − z 2 ) − 1 , d z arctan z = ( 1 + z 2 ) − 1 ,
d z a r c t a n h ( 1 / z ) = ( 1 − z 2 ) − 1 , d z arctan ( 1 / z ) = − ( 1 + z 2 ) − 1 . \mathrm{d}_z \mathrm{arctanh}\,(1/z) = (1 -z^2)^{-1} \;,
\qquad
\mathrm{d}_z \arctan(1/z) = -(1 +z^2)^{-1} \;.
d z a r c t a n h ( 1 / z ) = ( 1 − z 2 ) − 1 , d z arctan ( 1 / z ) = − ( 1 + z 2 ) − 1 .
Производные высших порядков:
d z r a r c t a n h z = ( r − 1 ) ! ⋅ ( 1 − z 2 ) − r ⋅ ∑ k C r 2 k + 1 ⋅ z r − 2 k − 1 \mathrm{d}_z^r \,\mathrm{arctanh}\, z =(r-1)!\cdot \bigl(1-z^2\bigr)^{-r}\cdot
\sum_k C_r^{2 k +1}\cdot z^{r -2 k -1}
d z r a r c t a n h z = ( r − 1 ) ! ⋅ ( 1 − z 2 ) − r ⋅ k ∑ C r 2 k + 1 ⋅ z r − 2 k − 1
= ( r − 1 ) ! ⋅ ( 1 − z 2 ) − r ⋅ ( C r 1 ⋅ z r − 1 + C r 3 ⋅ z r − 3 + . . . ) ; =(r-1)!\cdot \bigl(1-z^2\bigr)^{-r}\cdot
\Bigl(C_r^1\cdot z^{r-1} +C_r^3\cdot z^{r-3} +...\Bigr) \;;
= ( r − 1 ) ! ⋅ ( 1 − z 2 ) − r ⋅ ( C r 1 ⋅ z r − 1 + C r 3 ⋅ z r − 3 + . . . ) ;
d z r arctan z = ( r − 1 ) ! ⋅ ( 1 + z 2 ) − r ⋅ ∑ k ( − 1 ) r − k ⋅ C r 2 k − 1 ⋅ z r − 2 k + 1 \mathrm{d}_z^r \,\arctan z =(r-1)!\cdot \bigl(1+z^2\bigr)^{-r}\cdot
\sum_k (-1)^{r-k}\cdot C_r^{2 k -1}\cdot z^{r -2 k +1}
d z r arctan z = ( r − 1 ) ! ⋅ ( 1 + z 2 ) − r ⋅ k ∑ ( − 1 ) r − k ⋅ C r 2 k − 1 ⋅ z r − 2 k + 1
= ( r − 1 ) ! ⋅ ( 1 + z 2 ) − r ⋅ ( − 1 ) r ( − C r 1 ⋅ z r − 1 + C r 3 ⋅ z r − 3 − . . . ) . =(r-1)!\cdot \bigl(1+z^2\bigr)^{-r}\cdot
(-1)^r \Bigl(-C_r^1\cdot z^{r-1} +C_r^3\cdot z^{r-3} -...\Bigr) \;.
= ( r − 1 ) ! ⋅ ( 1 + z 2 ) − r ⋅ ( − 1 ) r ( − C r 1 ⋅ z r − 1 + C r 3 ⋅ z r − 3 − . . . ) .
d z a r c s i n h z = ( z 2 + 1 ) − 1 / 2 , d z arcsin z = ( 1 − z 2 ) − 1 / 2 , \mathrm{d}_z \mathrm{arcsinh}\, z = (z^2 +1)^{-1/2} \;,
\qquad
\mathrm{d}_z \arcsin z = (1 -z^2)^{-1/2} \;,
d z a r c s i n h z = ( z 2 + 1 ) − 1 / 2 , d z arcsin z = ( 1 − z 2 ) − 1 / 2 ,
d z a r c c o s h z = ( z 2 − 1 ) − 1 / 2 , d z arccos z = − ( 1 − z 2 ) − 1 / 2 . \mathrm{d}_z \mathrm{arccosh}\, z = (z^2 -1)^{-1/2} \;,
\qquad
\mathrm{d}_z \arccos z = -(1 -z^2)^{-1/2} \;.
d z a r c c o s h z = ( z 2 − 1 ) − 1 / 2 , d z arccos z = − ( 1 − z 2 ) − 1 / 2 .
a r c t a n h z = ∑ k = 0 ∞ 1 2 k + 1 ⋅ z 2 k + 1 = z + 1 3 ⋅ z 3 + 1 5 ⋅ z 5 + . . . ( ∣ z ∣ < 1 ) ;
\mathrm{arctanh}\, z = \sum_{k=0}^{\infty} \frac{1}{2k+1}\cdot z^{2k+1}
=z +\frac{1}{3}\cdot z^3 +\frac{1}{5}\cdot z^5 +...
\qquad (|z|<1) \;;
a r c t a n h z = k = 0 ∑ ∞ 2 k + 1 1 ⋅ z 2 k + 1 = z + 3 1 ⋅ z 3 + 5 1 ⋅ z 5 + . . . ( ∣ z ∣ < 1 ) ;
arctan z = ∑ k = 0 ∞ ( − 1 ) k 2 k + 1 ⋅ z 2 k + 1 = z − 1 3 ⋅ z 3 + 1 5 ⋅ z 5 − . . . ( ∣ z ∣ < 1 ) .
\arctan z = \sum_{k=0}^{\infty} \frac{ (-1)^k }{ 2k+1 }\cdot z^{2k+1}
=z -\frac{1}{3}\cdot z^3 +\frac{1}{5}\cdot z^5 -...
\qquad (|z|<1) \;.
arctan z = k = 0 ∑ ∞ 2 k + 1 ( − 1 ) k ⋅ z 2 k + 1 = z − 3 1 ⋅ z 3 + 5 1 ⋅ z 5 − . . . ( ∣ z ∣ < 1 ) .
Замечания, касающиеся вычисления значений функций с помощью рядов:
При вычислении функции arctan x \arctan x arctan x x > 0 x>0 x > 0
arctan x = 2 π / 4 − arctan ( 1 / x ) = 2 π / 8 − arctan ( ( 1 − x ) / ( 1 + x ) ) \arctan x = 2\pi/4 - \arctan(1/x) =2\pi/8 -\arctan\bigl((1-x)/(1+x)\bigr)
arctan x = 2 π / 4 − arctan ( 1 / x ) = 2 π / 8 − arctan ( ( 1 − x ) / ( 1 + x ) )
= 2 π / 16 − arctan ( ( 2 − 1 ) − x 1 + x ⋅ ( 2 − 1 ) ) . =2\pi/16 -\arctan\biggl(
\frac{\bigl(\sqrt{2} -1\bigr) -x }{ 1+x\cdot \bigl(\sqrt{2}-1\bigr) }\biggr) \;.
= 2 π / 1 6 − arctan ( 1 + x ⋅ ( 2 − 1 ) ( 2 − 1 ) − x ) .
При ∣ z ∣ < 1 |z|<1 ∣ z ∣ < 1
arcsin z = z ( 1 + ∑ k = 1 ∞ 1 ⋅ 3 ⋅ . . . ( 2 k − 1 ) 2 k ⋅ k ! ⋅ ( 2 k + 1 ) ⋅ z 2 k ) \arcsin z
= z\left(1 +\sum_{k=1}^{\infty}
\frac{1\cdot 3\cdot ...(2k-1)}{2^k \cdot k!\cdot (2 k +1)}\cdot z^{2 k}\right)
arcsin z = z ( 1 + k = 1 ∑ ∞ 2 k ⋅ k ! ⋅ ( 2 k + 1 ) 1 ⋅ 3 ⋅ . . . ( 2 k − 1 ) ⋅ z 2 k )
= z + ( 1 / 6 ) ⋅ z 3 + ( 3 / 40 ) ⋅ z 5 + ( 5 / 112 ) ⋅ z 7 + ( 35 / 1152 ) ⋅ z 9 + . . . . =z +(1/6)\cdot z^3 +(3/40)\cdot z^5
+(5/112)\cdot z^7 +(35/1152)\cdot z^9 +... \;.
= z + ( 1 / 6 ) ⋅ z 3 + ( 3 / 4 0 ) ⋅ z 5 + ( 5 / 1 1 2 ) ⋅ z 7 + ( 3 5 / 1 1 5 2 ) ⋅ z 9 + . . . .
a r c s i n h z = z ( 1 + ∑ k = 1 ∞ ( − 1 ) k 1 ⋅ 3 ⋅ . . . ( 2 k − 1 ) 2 k ⋅ k ! ⋅ ( 2 k + 1 ) ⋅ z 2 k ) \mathrm{arcsinh}\, z
=z\left(1 +\sum_{k=1}^{\infty} (-1)^k
\frac{ 1\cdot 3\cdot ...(2 k -1) }{ 2^k \cdot k!\cdot (2 k +1)}\cdot z^{2 k}\right)
a r c s i n h z = z ( 1 + k = 1 ∑ ∞ ( − 1 ) k 2 k ⋅ k ! ⋅ ( 2 k + 1 ) 1 ⋅ 3 ⋅ . . . ( 2 k − 1 ) ⋅ z 2 k )
= z − ( 1 / 6 ) ⋅ z 3 + ( 3 / 40 ) ⋅ z 5 − ( 5 / 112 ) ⋅ z 7 + ( 35 / 1152 ) ⋅ z 9 − . . . . =z -(1/6)\cdot z^3 +(3/40)\cdot z^5
-(5/112)\cdot z^7 +(35/1152)\cdot z^9 -... \;.
= z − ( 1 / 6 ) ⋅ z 3 + ( 3 / 4 0 ) ⋅ z 5 − ( 5 / 1 1 2 ) ⋅ z 7 + ( 3 5 / 1 1 5 2 ) ⋅ z 9 − . . . .
a) Для любого конечного z z z
a r c t a n h z = ∑ k = 0 N − 1 z 2 k + 1 2 k + 1 + ϱ N ( z ) ⋅ z 2 N + 1 2 N + 1 , \mathrm{arctanh}\, z =\sum_{k=0}^{N-1} \frac{ z^{2 k +1} }{2 k +1}
+\varrho_N(z)\cdot \frac{ z^{2 N +1} }{2 N +1} \;,
a r c t a n h z = k = 0 ∑ N − 1 2 k + 1 z 2 k + 1 + ϱ N ( z ) ⋅ 2 N + 1 z 2 N + 1 ,
где
ϱ ( z ) = 2 F 1 ( 1 , N + 1 / 2 ; N + 3 / 2 ; z 2 ) . \varrho(z) ={}_2 F_1\bigl(1,\, N+1/2;\, N+3/2;\, z^2\bigr) \;.
ϱ ( z ) = 2 F 1 ( 1 , N + 1 / 2 ; N + 3 / 2 ; z 2 ) .
При ∣ z ∣ < 1 |z|<1 ∣ z ∣ < 1
∣ ϱ N ( z ) ∣ < ( 1 − ∣ z ∣ 2 ) − 1 . |\varrho_N(z)|< \bigl(1-|z|^2\bigr)^{-1} \;.
∣ ϱ N ( z ) ∣ < ( 1 − ∣ z ∣ 2 ) − 1 .
b) Для любого конечного z z z
arcsin z = z + ∑ k = 1 N − 1 w k z 2 k + 1 + ϱ N ( z ) ⋅ w N z 2 N + 1 , \arcsin z =z +\sum_{k=1}^{N-1} w_k \,z^{2 k +1} +\varrho_N(z)\cdot w_N \,z^{2 N +1} \;,
arcsin z = z + k = 1 ∑ N − 1 w k z 2 k + 1 + ϱ N ( z ) ⋅ w N z 2 N + 1 ,
где
w k = 1 ⋅ 3 ⋅ . . . ( 2 k − 1 ) 2 k ⋅ k ! ⋅ ( 2 k + 1 ) , w_k =\frac{1\cdot 3\cdot ...(2k-1)}{2^k \cdot k!\cdot (2 k +1)} \;,
w k = 2 k ⋅ k ! ⋅ ( 2 k + 1 ) 1 ⋅ 3 ⋅ . . . ( 2 k − 1 ) ,
ϱ ( z ) = 3 F 2 ( 1 , N + 1 / 2 , N + 1 / 2 ; N + 1 , N + 3 / 2 ; z 2 ) . \varrho(z) ={}_3 F_2\bigl(1,\, N+1/2,\, N+1/2;\, N+1,\, N+3/2;\, z^2\bigr) \;.
ϱ ( z ) = 3 F 2 ( 1 , N + 1 / 2 , N + 1 / 2 ; N + 1 , N + 3 / 2 ; z 2 ) .
При ∣ z ∣ < 1 |z|<1 ∣ z ∣ < 1
∣ ϱ N ( z ) ∣ < ( 1 − ∣ z ∣ 2 ) − 1 . |\varrho_N(z)|< \bigl(1-|z|^2\bigr)^{-1} \;.
∣ ϱ N ( z ) ∣ < ( 1 − ∣ z ∣ 2 ) − 1 .
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