Inverse hyperbolic sine: Representations through equivalent functions

ArcSinh

$Mathematica Notation$

Elementary Functions

ArcSinh[z]

Representations through equivalent functions

With inverse function

Involving sinh^-1(sinh(z))

Involving sinh(sinh^-1(z))

With related functions

Involving log

Involving sin^-1

Involving sinh^-1(z)

Involving sinh^-1(z) and sin^-1(i z)

Involving sinh^-1(z) and sin^-1(1+2 z²)

Involving sinh^-1(z) and sin^-1((1-+ z)^1/2/2^1/2)

Involving sinh^-1(z) and sin^-1((1-i z)^1/2/2^1/2)

Involving sinh^-1(z) and sin^-1((-z²)^1/2)

Involving sinh^-1(z) and sin^-1((1+z²)^1/2)

Involving sinh^-1(z) and sin^-1(2 z (-z²-1)^1/2)

Involving sinh^-1(i z)

Involving sinh^-1(i z) and sin^-1(z)

Involving sinh^-1(z^1/2)

Involving sinh^-1(z^1/2) and sin^-1((-z)^1/2)

Involving sinh^-1(1/z^1/2)

Involving sinh^-1(1/z^1/2) and sin^-1(1/(-z)^1/2)

Involving sinh^-1(1/z^1/2) and sin^-1((-1/z)^1/2)

Involving sinh^-1((c z²)^1/2)

Involving sinh^-1((z₂)^1/2) and sin^-1(i z)

Involving sinh^-1((-z²)^1/2) and sin^-1(z)

Involving sinh^-1(a (b z^c)^m)

Involving sinh^-1(a (b z^c)^m) and sin^-1(i a b^m z^{m c})

Involving sinh^-1((z-1)^1/2)

Involving sinh^-1((z-1)^1/2) and sin^-1(z^1/2)

Involving sinh^-1((-1+c z)^1/2/2^1/2)

Involving sinh^-1((-1-z)^1/2/2^1/2) and sin^-1(z)

Involving sinh^-1((z-1)^1/2/2^1/2) and sin^-1(z)

Involving sinh^-1((1-z)^1/2/z^1/2)

Involving sinh^-1((1-z)^1/2/z^1/2) and sin^-1(1/z^1/2)

Involving sinh^-1((z-1)^1/2/(-z)^1/2)

Involving sinh^-1((z-1)^1/2/(-z)^1/2) and sin^-1(1/z^1/2)

Involving sinh^-1(1-z/z^1/2)

Involving sinh^-1(1-z/z^1/2) and sin^-1(1/z^1/2)

Involving sinh^-1((a-z)^1/2/(2z)^1/2)

Involving sinh^-1((-z-1)^1/2/(2 z)^1/2) and sin^-1(1/z)

Involving sinh^-1((1-z)^1/2/(2 z)^1/2) and sin^-1(1/z)

Involving sinh^-1((z-a)^1/2/(-2z)^1/2)

Involving sinh^-1((z+1)^1/2/(-2 z)^1/2) and sin^-1(1/z)

Involving sinh^-1((z-1)^1/2/(-2 z)^1/2) and sin^-1(1/z)

Involving sinh^-1(a-z/2z^1/2)

Involving sinh^-1(-z-1/2 z^1/2) and sin^-1(1/z)

Involving sinh^-1(1-z/2z^1/2) and sin^-1(1/z)

Involving sinh^-1((a z^c-1)^1/2)

Involving sinh^-1((a z^c-1)^1/2) and sin^-1(a^1/2 z^c/2)

Involving sinh^-1((1-z²)^1/2/z)

Involving sinh^-1((1-z²)^1/2/z) and sin^-1(1/z)

Involving sinh^-1((1-z²)^1/2/(z₂)^1/2)

Involving sinh^-1((1-z²)^1/2/(z₂)^1/2) and sin^-1(1/z)

Involving sinh^-1((z²-1)^1/2/(-z²)^1/2)

Involving sinh^-1((z²-1)^1/2/(-z²)^1/2) and sin^-1(1/z)

Involving sinh^-1(1-z²/z²^1/2)

Involving sinh^-1(1-z²/z²^1/2) and sin^-1(1/z)

Involving sinh^-1(2 z (z²-1)^1/2)

Involving sinh^-1(2 z (z²-1)^1/2) and sin^-1(z)

Involving sinh^-1(2 (1-z²)^1/2/z²)

Involving sinh^-1(2 (1-z²)^1/2/z²) and sin^-1(1/z)

Involving sinh^-1((((1-z²)^1/2-1)/2)^1/2)

Involving sinh^-1((((1-z²)^1/2-1)/2)^1/2) and sin^-1(z)

Involving sinh^-1(z ((1-z²)^1/2-1)^1/2/(2z²)^1/2)

Involving sinh^-1(z ((1-z²)^1/2-1)^1/2/(2z²)^1/2) and sin^-1(z)

Involving sinh^-1(z (((1-z²)^1/2-1)/(2z²))^1/2)

Involving sinh^-1(z (((1-z²)^1/2-1)/(2z²))^1/2) and sin^-1(z)

Involving sinh^-1(((z²-1)^1/2-z)^1/2/(2z)^1/2)

Involving sinh^-1(((z²-1)^1/2-z)^1/2/(2z)^1/2) and sin^-1(1/z)

Involving sinh^-1((((z²-1)^1/2-z)/(2z))^1/2)

Involving sinh^-1((((z²-1)^1/2-z)/(2z))^1/2) and sin^-1(1/z)

Involving cos^-1

Involving sinh^-1(z)

Involving sinh^-1(z) and cos^-1(i z)

Involving sinh^-1(i z) and cos^-1(z)

Involving sinh^-1(z) and cos^-1(1+2 z²)

Involving sinh^-1(z) and cos^-1((1+i z)^1/2/2^1/2)

Involving sinh^-1(z) and cos^-1((1-i z)^1/2/2^1/2)

Involving sinh^-1(z) and cos^-1((-z²)^1/2)

Involving sinh^-1(z) and cos^-1((1+z²)^1/2)

Involving sinh^-1(z) and cos^-1(2 z (-z²-1)^1/2)

Involving sinh^-1(z^1/2)

Involving sinh^-1(z^1/2) and cos^-1((-z)^1/2)

Involving sinh^-1(1/z^1/2)

Involving sinh^-1(1/z^1/2) and cos^-1(1/(-z)^1/2)

Involving sinh^-1(1/z^1/2) and cos^-1((-1/z)^1/2)

Involving sinh^-1((c z²)^1/2)

Involving sinh^-1((z₂)^1/2) and cos^-1(i z)

Involving sinh^-1((-z²)^1/2) and cos^-1(z)

Involving sinh^-1(a (b z^c)^m)

Involving sinh^-1(a (b z^c)^m) and cos^-1(i a b^m z^{m c})

Involving sinh^-1((z-1)^1/2)

Involving sinh^-1((z-1)^1/2) and cos^-1(z^1/2)

Involving sinh^-1((-1+c z)^1/2/2^1/2)

Involving sinh^-1((-1-z)^1/2/2^1/2) and cos^-1(z)

Involving sinh^-1((z-1)^1/2/2^1/2) and cos^-1(z)

Involving sinh^-1((1-z)^1/2/z^1/2)

Involving sinh^-1((1-z)^1/2/z^1/2) and cos^-1(1/z^1/2)

Involving sinh^-1((z-1)^1/2/(-z)^1/2)

Involving sinh^-1((z-1)^1/2/(-z)^1/2) and cos^-1(1/z^1/2)

Involving sinh^-1(1-z/z^1/2)

Involving sinh^-1(1-z/z^1/2) and cos^-1(1/z^1/2)

Involving sinh^-1((a-z)^1/2/(2z)^1/2)

Involving sinh^-1((-z-1)^1/2/(2 z)^1/2) and cos^-1(1/z)

Involving sinh^-1((1-z)^1/2/(2 z)^1/2) and cos^-1(1/z)

Involving sinh^-1((z-a)^1/2/(-2z)^1/2)

Involving sinh^-1((z+1)^1/2/(-2 z)^1/2) and cos^-1(1/z)

Involving sinh^-1((z-1)^1/2/(-2 z)^1/2) and cos^-1(1/z)

Involving sinh^-1(a-z/2z^1/2)

Involving sinh^-1(-z-1/2 z^1/2) and cos^-1(1/z)

Involving sinh^-1(1-z/2z^1/2) and cos^-1(1/z)

Involving sinh^-1((a z^c-1)^1/2)

Involving sinh^-1((a z^c-1)^1/2) and cos^-1(a^1/2 z^c/2)

Involving sinh^-1((1-z²)^1/2/z)

Involving sinh^-1((1-z²)^1/2/z) and cos^-1(1/z)

Involving sinh^-1((1-z²)^1/2/(z₂)^1/2)

Involving sinh^-1((1-z²)^1/2/(z₂)^1/2) and cos^-1(1/z)

Involving sinh^-1((z²-1)^1/2/(-z²)^1/2)

Involving sinh^-1((z²-1)^1/2/(-z²)^1/2) and cos^-1(1/z)

Involving sinh^-1(1-z²/z²^1/2)

Involving sinh^-1(1-z²/z²^1/2) and cos^-1(1/z)

Involving sinh^-1(2 z (z²-1)^1/2)

Involving sinh^-1(2 z (z²-1)^1/2) and cos^-1(z)

Involving sinh^-1(2 (1-z²)^1/2/z²)

Involving sinh^-1(2 (1-z²)^1/2/z²) and cos^-1(1/z)

Involving sinh^-1((((1-z²)^1/2-1)/2)^1/2)

Involving sinh^-1((((1-z²)^1/2-1)/2)^1/2) and cos^-1(z)

Involving sinh^-1(z ((1-z²)^1/2-1)^1/2/(2z²)^1/2)

Involving sinh^-1(z ((1-z²)^1/2-1)^1/2/(2z²)^1/2) and cos^-1(z)

Involving sinh^-1(z (((1-z²)^1/2-1)/(2z²))^1/2)

Involving sinh^-1(z (((1-z²)^1/2-1)/(2z²))^1/2) and cos^-1(z)

Involving sinh^-1(((z²-1)^1/2-z)^1/2/(2z)^1/2)

Involving sinh^-1(((z²-1)^1/2-z)^1/2/(2z)^1/2) and cos^-1(1/z)

Involving sinh^-1((((z²-1)^1/2-z)/(2z))^1/2)

Involving sinh^-1((((z²-1)^1/2-z)/(2z))^1/2) and cos^-1(1/z)

Involving tan^-1

Involving sinh^-1(z)

Involving sinh^-1(z) and tan^-1((-1-z²)^1/2/z)

Involving sinh^-1(z) and tan^-1((-z²-1)^1/2/(z₂)^1/2)

Involving sinh^-1(z) and tan^-1((1+z²)^1/2/(-z²)^1/2)

Involving sinh^-1(z) and tan^-1((-z²+1/z²)^1/2)

Involving sinh^-1(z) and tan^-1(z/(-z²-1)^1/2)

Involving sinh^-1(z) and tan^-1((-z²)^1/2/(z²+1)^1/2)

Involving sinh^-1(z) and tan^-1((z₂)^1/2/(-z²-1)^1/2)

Involving sinh^-1(z) and tan^-1((-z²/z²+1)^1/2)

Involving sinh^-1(z) and tan^-1(2 z (-z²-1)^1/2/1+2 z²)

Involving sinh^-1(z) and tan^-1(1+2z²/2 z (-1-z²)^1/2)

Involving sinh^-1(z) and tan^-1((-i z-1)^1/2/(i z-1)^1/2)

Involving sinh^-1(z) and tan^-1((1+i z)^1/2/(1-i z)^1/2)

Involving sinh^-1(z) and tan^-1((z-i)^1/2/(-z-i)^1/2)

Involving sinh^-1(z) and tan^-1((i-z)^1/2/(i+z)^1/2)

Involving sinh^-1(z) and tan^-1(i-z/i+z^1/2)

Involving sinh^-1(z) and tan^-1((i z-1)^1/2/(-i z-1)^1/2)

Involving sinh^-1(z) and tan^-1((1-i z)^1/2/(1+i z)^1/2)

Involving sinh^-1(z) and tan^-1((-z-i)^1/2/(z-i)^1/2)

Involving sinh^-1(z) and tan^-1((i+z)^1/2/(i-z)^1/2)

Involving sinh^-1(z) and tan^-1(i+z/i-z^1/2)

Involving sinh^-1(2 z/1-z²)

Involving sinh^-1(2 z/1-z²) and tan^-1(i z)

Involving sinh^-1(2 z/1-z²) and tan^-1(i/z)

Involving sinh^-1(2 z/1-z²) and tan^-1(i z^r)

Involving sinh^-1(2 z/z²-1)

Involving sinh^-1(2 z/z²-1) and tan^-1(i z)

Involving sinh^-1(2 z/z²-1) and tan^-1(i/z)

Involving sinh^-1(2 z/z²-1) and tan^-1(i z^r)

Involving sinh^-1(1/(-z-1)^1/2)

Involving sinh^-1(1/(-z-1)^1/2) and tan^-1(z^1/2)

Involving sinh^-1(1/(-z-1)^1/2) and tan^-1(1/z^1/2)

Involving sinh^-1(1/-1-z^1/2)

Involving sinh^-1(1/-1-z^1/2) and tan^-1(z^1/2)

Involving sinh^-1(1/-1-z^1/2) and tan^-1(1/z^1/2)

Involving sinh^-1((-z)^1/2/(1+z)^1/2)

Involving sinh^-1((-z)^1/2/(1+z)^1/2) and tan^-1(z^1/2)

Involving sinh^-1((-z)^1/2/(1+z)^1/2) and tan^-1(1/z^1/2)

Involving sinh^-1(z^1/2/(-1-z)^1/2)

Involving sinh^-1(z^1/2/(-1-z)^1/2) and tan^-1(z^1/2)

Involving sinh^-1(z^1/2/(-1-z)^1/2) and tan^-1(1/z^1/2)

Involving sinh^-1(z/-1-z^1/2)

Involving sinh^-1(z/-1-z^1/2) and tan^-1(z^1/2)

Involving sinh^-1(z/-1-z^1/2) and tan^-1(1/z^1/2)

Involving sinh^-1(1/(-1-z²)^1/2)

Involving sinh^-1(1/(-1-z²)^1/2) and tan^-1(z)

Involving sinh^-1(1/(-1-z²)^1/2) and tan^-1(1/z)

Involving sinh^-1(1/-1-z²^1/2)

Involving sinh^-1(1/-1-z²^1/2) and tan^-1(z)

Involving sinh^-1(1/-1-z²^1/2) and tan^-1(1/z)

Involving sinh^-1(z/(-1-z²)^1/2)

Involving sinh^-1(z/(-1-z²)^1/2) and tan^-1(z)

Involving sinh^-1(z/(-1-z²)^1/2) and tan^-1(1/z)

Involving sinh^-1((z₂)^1/2/(-1-z²)^1/2)

Involving sinh^-1((z₂)^1/2/(-1-z²)^1/2) and tan^-1(z)

Involving sinh^-1((z₂)^1/2/(-1-z²)^1/2) and tan^-1(1/z)

Involving sinh^-1((-z²)^1/2/(1+z²)^1/2)

Involving sinh^-1((-z²)^1/2/(1+z²)^1/2) and tan^-1(z)

Involving sinh^-1((-z²)^1/2/(1+z²)^1/2) and tan^-1(1/z)

Involving sinh^-1(z²/-z²-1^1/2)

Involving sinh^-1(z²/-z²-1^1/2) and tan^-1(z)

Involving sinh^-1(z²/-z²-1^1/2) and tan^-1(1/z)

Involving sinh^-1((1-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4))

Involving sinh^-1((1-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4)) and tan^-1(z)

Involving sinh^-1((1-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4)) and tan^-1(1/z)

Involving sinh^-1((z-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4))

Involving sinh^-1((z-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4)) and tan^-1(z)

Involving sinh^-1((z-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4)) and tan^-1(1/z)

Involving sinh^-1(((1-(1+z²)^1/2)/(2(1+z²)^1/2))^1/2)

Involving sinh^-1(((1-(1+z²)^1/2)/(2(1+z²)^1/2))^1/2) and tan^-1(z)

Involving sinh^-1(((1-(1+z²)^1/2)/(2(1+z²)^1/2))^1/2) and tan^-1(1/z)

Involving sinh^-1(((z-(1+z²)^1/2)/(2(1+z²)^1/2))^1/2)

Involving sinh^-1(((z-(1+z²)^1/2)/(2(1+z²)^1/2))^1/2) and tan^-1(z)

Involving sinh^-1(((z-(1+z²)^1/2)/(2(1+z²)^1/2))^1/2) and tan^-1(1/z)

Involving cot^-1

Involving sinh^-1(z)

Involving sinh^-1(z) and cot^-1((-z²-1)^1/2/z)

Involving sinh^-1(z) and cot^-1((-z²-1)^1/2/(z₂)^1/2)

Involving sinh^-1(z) and cot^-1((z²+1)^1/2/(-z²)^1/2)

Involving sinh^-1(z) and cot^-1((-z²+1/z²)^1/2)

Involving sinh^-1(z) and cot^-1(z/(-1-z²)^1/2)

Involving sinh^-1(z) and cot^-1((-z²)^1/2/(1+z²)^1/2)

Involving sinh^-1(z) and cot^-1((z₂)^1/2/(-z²-1)^1/2)

Involving sinh^-1(z) and cot^-1((-z²/z²+1)^1/2)

Involving sinh^-1(z) and cot^-1(2 z (-1-z²)^1/2/1+2z²)

Involving sinh^-1(z) and cot^-1(1+2 z²/2 z (-z²-1)^1/2)

Involving sinh^-1(z) and cot^-1((-i z-1)^1/2/(i z-1)^1/2)

Involving sinh^-1(z) and cot^-1((1+i z)^1/2/(1-i z)^1/2)

Involving sinh^-1(z) and cot^-1((z-i)^1/2/(-z-i)^1/2)

Involving sinh^-1(z) and cot^-1((i-z)^1/2/(i+z)^1/2)

Involving sinh^-1(z) and cot^-1(i-z/i+z^1/2)

Involving sinh^-1(z) and cot^-1((i z-1)^1/2/(-i z-1)^1/2)

Involving sinh^-1(z) and cot^-1((1-i z)^1/2/(1+i z)^1/2)

Involving sinh^-1(z) and cot^-1((-z-i)^1/2/(z-i)^1/2)

Involving sinh^-1(z) and cot^-1((i+z)^1/2/(i-z)^1/2)

Involving sinh^-1(z) and cot^-1(i+z/i-z^1/2)

Involving sinh^-1(2 z/1-z²)

Involving sinh^-1(2 z/1-z²) and cot^-1(i z)

Involving sinh^-1(2 z/1-z²) and cot^-1(i/z)

Involving sinh^-1(2 z/1-z²) and cot^-1(i z^r)

Involving sinh^-1(2 z/z²-1)

Involving sinh^-1(2 z/z²-1) and cot^-1(i z)

Involving sinh^-1(2 z/z²-1) and cot^-1(i/z)

Involving sinh^-1(2 z/z²-1) and cot^-1(i z^r)

Involving sinh^-1(1/(-z-1)^1/2)

Involving sinh^-1(1/(-z-1)^1/2) and cot^-1(z^1/2)

Involving sinh^-1(1/(-z-1)^1/2) and cot^-1(1/z^1/2)

Involving sinh^-1(1/-1-z^1/2)

Involving sinh^-1(1/-1-z^1/2) and cot^-1(z^1/2)

Involving sinh^-1(1/-1-z^1/2) and cot^-1(1/z^1/2)

Involving sinh^-1((-z)^1/2/(1+z)^1/2)

Involving sinh^-1((-z)^1/2/(1+z)^1/2) and cot^-1(z^1/2)

Involving sinh^-1((-z)^1/2/(1+z)^1/2) and cot^-1(1/z^1/2)

Involving sinh^-1(z^1/2/(-1-z)^1/2)

Involving sinh^-1(z^1/2/(-1-z)^1/2) and cot^-1(z^1/2)

Involving sinh^-1(z^1/2/(-1-z)^1/2) and cot^-1(1/z^1/2)

Involving sinh^-1(z/-1-z^1/2)

Involving sinh^-1(z/-1-z^1/2) and cot^-1(z^1/2)

Involving sinh^-1(z/-1-z^1/2) and cot^-1(1/z^1/2)

Involving sinh^-1(1/(-1-z²)^1/2)

Involving sinh^-1(1/(-1-z²)^1/2) and cot^-1(z)

Involving sinh^-1(1/(-1-z²)^1/2) and cot^-1(1/z)

Involving sinh^-1(1/-1-z²^1/2)

Involving sinh^-1(1/-1-z²^1/2) and cot^-1(z)

Involving sinh^-1(1/-1-z²^1/2) and cot^-1(1/z)

Involving sinh^-1(z/(-1-z²)^1/2)

Involving sinh^-1(z/(-1-z²)^1/2) and cot^-1(z)

Involving sinh^-1(z/(-1-z²)^1/2) and cot^-1(1/z)

Involving sinh^-1((z₂)^1/2/(-1-z²)^1/2)

Involving sinh^-1((z₂)^1/2/(-1-z²)^1/2) and cot^-1(z)

Involving sinh^-1((z₂)^1/2/(-1-z²)^1/2) and cot^-1(1/z)

Involving sinh^-1((-z²)^1/2/(1+z²)^1/2)

Involving sinh^-1((-z²)^1/2/(1+z²)^1/2) and cot^-1(z)

Involving sinh^-1((-z²)^1/2/(1+z²)^1/2) and cot^-1(1/z)

Involving sinh^-1(z²/-z²-1^1/2)

Involving sinh^-1(z²/-z²-1^1/2) and cot^-1(z)

Involving sinh^-1(z²/-z²-1^1/2) and cot^-1(1/z)

Involving sinh^-1((1-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4))

Involving sinh^-1((1-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4)) and cot^-1(z)

Involving sinh^-1((1-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4)) and cot^-1(1/z)

Involving sinh^-1((z-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4))

Involving sinh^-1((z-(1+z²)^1/2)^1/2/(2^1/2 (1+z²)^1/4)) and cot^-1(z)