To find logarithm of a complex number

AlbertEinstein

#2

Hi,
Well can anyone tell me how to find the natural logarithm of a complex number p + iq.
Also please tell me how to convert it into logarithm to the base 10.
An external link to a webpage (where all the details are given) will be appreciated.

Tide

Write the complex number in polar form and it should be evident how to find the logarithm.

To convert the logarithm to base 10 use log z = ln z/ln 10.

HallsofIvy

Of course, since the argument of a complex number (the angle part of the polar form) can have any multiple of [itex]2\pi[/itex] added to it, the log function is, like most complex functions, multi-valued.

mathwonk

the defn of ln(z) is the integral of the differential dw/w along a path from w = 1 to w = z, but not passing through w=0. the ambiguity is in the choice of that path.

if you transform it to r,theta coordinates I am guessing that differential becomes dr/r + i dtheta. since dr/r has a primitive, namely it equals dln|r|, which is dwefined everywhere except at 0, that part of the integral is not dependent on the path.

but the other part idtheta, does depend on how much angle is swept out wrt the origin by the whole path.


so the real part of the log of z is just the log of the absolute value |z|, but the complex part equals i times some determination of the angle arg(z).

I didn't really calculate this out just now, but this is probably very close to correct, as this is one of my long time favorite objects. I never understood complex functions or logs until I saw it explained roughly this way in Courant's calculus book (where else?).

mathwonk

thats why 3^(ipi) = -1, i.e. minus one has absolute value 1 so real log zero, but angle pi, so complex part ipi.

and also why e^(2ipi) = 1, since one determination of the angle of 1 is 2pi.

Kaan

I remember something like

Ln(z) = Ln(|z|) + iArg(z)

where |z| is the amplitude
and arg(z) is the angle in radians between the point z and x axis.

HallsofIvy

And, again, since the angle Arg(z) + [itex]2kpi[/itex], for any integer i, gives the same point, ln(z)= ln(|z|)+ i Arg(z)+ [itex]2ki\pi[/itex]

rbj

i don't really want to start anything, wonk, but while i believe that is true, is it the definition?

i didn't even think that was the definition for the real logarithm

[tex] \log(x) \equiv \int_1^x \frac{1}{u} du [/tex]

i thought that the definition had more to do with

[tex] \log(x y) = \log(x) + \log(y) [/tex]

that's what defines some function as a logarithm.

then it can be shown that for the function:

[tex] f_A(x) \equiv \int_1^x \frac{A}{u} du [/tex]

[tex] f_A(xy) = \int_1^{xy} \frac{A}{u} du = \int_1^{x} \frac{A}{u} du + \int_x^{xy} \frac{A}{u} du = \int_1^{x} \frac{A}{u} du + \int_1^{y} \frac{A}{u} du = f_A(x) + f_A(y)[/tex]

so we can say that f_A is logarithmic. and the natural logarithm was such that numerator A is 1.

isn't this a difference between what is definition and what is a resulting property?

Vid

They're both perfectly valid definitions. Some properties are easier to prove using one choice of definition over the other.
An example is this PDF using an alternate definition of the exp(x), and deriving some specific results from that.
http://www.math.lsu.edu/~mcgehee/Exp.pdf

HallsofIvy

What makes you think there is such a thing as such a thing as "the" definition of ln(x)?
Any function can have many different equivalent definitions. though it is a matter of taste, I would think that a definition that gives a direct formula ([itex]ln(x)= \int_1^x dt/t[/itex]) is better than a definition that says the fnction has such and such properties (ln(x) is the function f such that f(xy)= f(x)+ f(y) and f(1)= 0)/

rbj

i guess i normally think that a definition comes first, chronologically in the history or pedagogy.

in that sense, i think of logarithms as the generalized exponent of some given base (which is "e" if it's a "natural logarithm" - why e takes on the value that it does has to do with that integral, that wonk was using as a defining expression).

i didn't like it, but in my old calc book (Seeley, and i generally really like the book), they simple defined the natural exponential of a complex number as

[tex] e^z = e^{\mathrm{Re}(z)} \cos\left(\mathrm{Im}(z)\right) \ + \ i e^{\mathrm{Re}(z)} \sin\left(\mathrm{Im}(z)\right) [/tex]

and then stated that the definition was a good one since it reverted to the previous definition of e^x for a real argument (when Im(z)=0) and satisfied e^z+w = e^z e^w for any complex z and w. for me, that was an unsatisfying "definition".

BCox

Hello:

So if my function was

LN [ -exp(cx)/sin(d) - i ] = ?

Then, I would decompose it to real and imaginary parts?

The larger problem is

LN [ -exp(cx)/sin(d) - i ] - LN [ -exp(cx)/sin(d) + i ]

where c and d are constants.

???

HallsofIvy

Assuming that -exp(cx)/sin(d) is real, that is of the form ln(a- i) with a= -exp(cx)/sin(d). Rewrite a- i in "polar form", [itex]re^{i\theta}[/itex] with [itex]r= \sqrt{exp(2cx)/sin^2(d)+ 1}[/itex][itex]= \sqrt{exp(2cx)+ sin^2(d)}/sin(d)[/itex] and [itex]\theta= arctan(sin(d)/exp(cx))[/itex]. Then [itex]ln(a- i)= ln(r)+ \theta i+ 2k\pi i[/itex] or
[tex]ln(-exp(cx)/sin(d)- i)= \frac{1}{2}ln(exp(2cx)+ sin^2(d))- ln(sin(d))+ arctan(sin(d)/exp(cx))i+ 2k\pi i[/tex]

BCox

Hmm... perhaps a rephrase of the question is more appropriate.

I want to take the limit of the following function as x -> infinity

(1/d) * arcot [ - exp(cx) / sin(d) ]

where d is (-pi,0)

Now a prior condition would stipulate that the above has to go to zero as x tends to infinity. But I am more interested at the rate at which the function goes to zero as x -> infinity.

For one case as d->0, the first term tends to infinity. But I have another condition telling me that the whole function must approach zero; so that the second term arcot() must approach zero at a faster rate than 1/d. But analytically, what is the rate that

(1/d) * arcot [ - exp(cx) / sin(d) ] ~ ? as x->infinity

clustro

I'm not really a "mathematician", but for what its worth, you could put the complex number into a power series and hopefully try to see what it converges too...no? :( ??

HallsofIvy

?? A power series is necessarily a function. How do you put a number in a power series.

BCox

right. It's tricky