Ln of a complex number

[Albert Chan]

(03-02-2024 09:30 AM)hp-zl Wrote:  re(ln(a+i*b)), im(ln(a+i*b))

Cas> parts(z) := [re(z), im(z)]
Cas> m := parts(ln(z));      // setting with complex off

[ln(abs(z)), 1/2*π*(-sign(z)+1)]

Cas> m(z=e), m(z=-e), m(z=0)

[1,0], [1,π], [-∞, 1/2*π]

Except for weird phase angle for ln(0), this looks OK

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Cas> m := parts(ln(z));      // setting with complex on

[ln(abs(z)), π*sign(im(z)) + atan(im(z)/re(z))]      // ?

HP Prime has a bug on imag part, using atan instead of atan2
Correct ln(z) imag part should be arg(z) = atan2(im(z), re(z))

Above is only correct for re(z) < 0

Cas> m(z=1+i)

[1/2*ln(2), 5/4*π]      // ?