erfi, erfw (w(z)) functions

[Albert Chan]

A mnemonic I find useful
swap -1 and i, and swap the function (erf ↔ erfi, sin ↔ sinh, cos ↔ cosh, tan ↔ tanh)

Code:

erf (-x) = - erf(x)    → erf (ix) = i erfi(x)
erfi(-x) = - erfi(x)   → erfi(ix) = i erf(x)

sin (-x) = - sin(x)    → sin (ix) = i sinh(x)
sinh(-x) = - sinh(x)   → sinh(ix) = i sin(x)

cos (-x) =   cos(x)    → cos (ix) =   cosh(x)
cosh(-x) =   cosh(x)   → cosh(ix) =   cos(x)

tan (-x) = - tan(x)    → tan (ix) = i tanh(x)
tanh(-x) = - tanh(x)   → tanh(ix) = i tan(x)

Update: we could replace i by 1/i, and get a new set of identities

cosh(x) = cos(i*x) = cos(-i*x) = cos(x/i)
atan(x) = atanh(ix)/i = atanh(-ix)/(-i) = atanh(x/i)*i
...

Update2: A better mnemonic maybe to turn function even

sin(x)/x = sinh(±i*x)/(±i*x) = 1 − x^2/6 + x^4/120 − ...
sinh(x)/x = sin(±i*x)/(±i*x) = 1 + x^2/6 + x^4/120 + ...