Quick help with complex numbers

[mizzl]

I wonder if anyone here knows this...

What's ln(sqrt(3) + i)?
In my calculation I do the following
|sqrt(3) + i| = sqrt(3 + 1) = 2
arg(sqrt(3)+i) = atan(1/sqrt(3)) = 1/3pi
so ln(sqrt(3) + i) =
ln(2e^((1/3)*pi*i)) =
ln(2) + ln(e^((1/3)*pi*i)) =
ln(2) + (1/3)*pi*i

Is this true?

[orcus116]

Is it equal to anything?

[mizzl]

no, not really

[orcus116]

Can you do:

e^ln(<equation>)

to get the equation out? I'm pretty sure you can do the opposite but I haven't done math like this in awhile.

[Elaitch]

https://www.wolframalpha.com/input/?i=is+ln%28sqrt%283%29%2Bi%29+%3D+ln%282%29+%2B+%281%2F3%29*pi*i+%3F

Apparently not.

[Jamesman42]

What is this for?   Complex analysis?

I don't think I'v encountered something like this before.

[orcus116]

Don't you just solve these two in terms of polar coordinates?

[Jamesman42]

Still unsure of what is going on exactly, but wolfram gave this: look under "Alternate forms"

https://www.wolframalpha.com/input/?i=ln%28sqrt%283%29+%2B+i%29&x=3&y=3

Looks like the term with pi is off by a factor of 1/2

[ddtonfire]

z = √3 + i
lnz = ln|z| + iargz
ln|z| = ln(sqrt(3 + 1)) = ln(2)
argz = π/6, from:

√3 + i = 2(cosθ + isinθ)
equate imaginary parts:
sinθ = 1/2, θ = π/6

so

ln(2) + i(π/6)

Hope that helps.

[mizzl]

z = √3 + i
lnz = ln|z| + iargz
ln|z| = ln(sqrt(3 + 1)) = ln(2)
argz = π/6, from:

√3 + i = 2(cosθ + isinθ)
equate imaginary parts:
sinθ = 1/2, θ = π/6

so

ln(2) + i(π/6)

Hope that helps.

Thanks! That solved it! Wonder what I did wrong though..

[ddtonfire]

arg(sqrt(3)+i) = atan(1/sqrt(3)) = 1/3pi

atan(1/sqrt(3)) = 1/6pi, not 1/3pi