4. arccot z = 1 2iln( iz − 1 iz + 1) = i arccoth(iz)

131
2.7 Inverse Trigonometric and Hyperbolic Functions
1
1 + iz
1
3. arctan z =
ln
= arctanh(i z)
2i
1 − iz
i
iz − 1
1
ln
= i arccoth(i z)
4. arccot z =
2i
iz + 1
5. arcsinh z = ln(z +
z 2 + 1) =
1
arcsin(i z)
i
6. arccosh z = ln(z + z 2 − 1) = i arccos z
1+z
1
1
= arctan(i z)
7. arctanh z = ln
2
1−z
i
z+1
1
1
= arccot(−i z)
8. arccoth z = ln
2
z−1
i
2.7.2.3 Relationships between different inverse trigonometric functions.
π
1. arcsin x + arccos x =
2
π
2. arctg x + arcctg x =
2
2.7.2.4
1. arcsin x = arccos 1 − x 2
= − arccos 1 − x 2
[0 ≤ x ≤ 1]
[−1 ≤ x ≤ 0]
x
[x 2 < 1]
2. arcsin x = arctan √
1 − x2
√
1 − x2
[0 < x ≤ 1]
3. arcsin x = arccot
x
√
1 − x2
−π
[−1 ≤ x < 0]
= arccot
x
[0 ≤ x ≤ 1]
4. arccos x = arcsin 1 − x 2
2
[−1 ≤ x ≤ 0]
= π − arcsin 1 − x
√
1 − x2
[0 < x ≤ 1]
5. arccos x = arctan
x
√
1 − x2
[−1 ≤ x < 0]
= π + arccot
x
x
6. arccos x = arccot √
1 − x2
[−1 ≤ x < 1]