But You Can't Do That

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

#math #brithemathguy #logarithm

[BriTheMathGuy]

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[angelmendez-rivera]

In practice, the complex logarithm is used when studying Riemann surfaces, which are important when doing contour integration, and it also gets used when studying certain geometries. However, mathematicians almost never work with evaluating the complex logarithm for individual complex numbers, precisely because of its ill-defined nature. You cannot do functional analysis or arithmetic with non-functions. So I think one may be justified in saying that ln(–3) is undefined, even if we use complex numbers, because of this issue.

[der.Schtefan]

Using the word "imaginary" for them was the worst mistake ever.

[JaybeePenaflor]

This is one of my favorite concepts in complex analysis!

The natural logarithm function in complex analysis is multi-valued, as shown in the video. Suppose we have z = e^(iθ). Suppose we start at 0, and then rotate counterclockwise until you reach 2π, taking ln z along the way. At 2π, the function has now taken a new value. This is similar to rotating along a spiral vertically: you will never reach the same value of ln z even if you returned to the original position for z.

Thus, to prevent this from happening, we define the limit for which we can choose values for θ. This is called defining a principal branch of the logarithmic function. We usually do a branch cut along the negative real axis, thus limiting the imaginary part between −π and π. The resulting function is typically indicated by a capital first letter, Ln z, to distinguish it from the multi-valued, ln z.

[thssc]

By Euler's Identity,
e^πi = -1.
Take natural log of both sides,
We get πi = ln(-1).

Now let us obtain ln(-x).
We know, - x = -1 * x = e^πi * x.
Therefore,
-x = e^πi * x.

Take natural log of both sides,
We get ln(-x) = πi + ln(x)
Put x = 3,

ln(-3) = πi + ln(3)

[telmanm]

well, e^iπ+1=0 rerrange you get, e^iπ=-1 so, e^x+1=0 solving for x you get ln(-1) and, e^iπ=-1, so it would be:ln(e^iπ) so, ln(e)=1 then, it equals iπ:
e^iπ+1=0 and ln(-1)=iπ.

[canaDavid]

There are two problems with this: one is that it isn't a function from C to C, it's a function from C to C[k], as the output depends on some integer k. To make it from C to C, one could decide to take the smallest absolute angle, ie ln(i) = ipi/2 etc. But this does not work for negative numbers, as both ipi and -ipi work. And choosing one of them would not work, as it would break the nontrivial field automorphism C->C: i -> -i.

[babai_]

I love the length of this video, works really well for this video too 😂

[idoabramovich]

You can say that:

I know it's false but it's nice one

[Thitadhammo]

Well, that seems obvious enough. Whenever YouTube mathematicians post something like 'they told you you can't do this', it's frequently the case complex numbers save the day. I'm disappointed, though. Pi didn't feature centrally while the video is 3:14 minutes long. That ought to be a sin.

[randomsupermario]

Don't we restrict theta to be between 0 and 2pi? This would make it well-defined!

[erpaninozzo]

the duration of the video is 3:14 minutes, I don't think I need to add more

[Nikioko]

Logarithms of negative numbers are like square roots of negative numbers. And logarithms with base 1 are like division by 0.

[lakshya]

Ohh
Its not that difficult to understand, its just that I've never thought of it
Great work

[michaellarson]

To fix that problem of getting infinitely many answers, you just use the rule that the absolute value of theta must be less than or equal to pi.

[joseph_soseph]

The way I think about it is by splitting the logarithm
ln(ab) = ln(a) + ln(b)
=> ln(-3) = ln(3) + ln(-1)
With Euler's identity, I find:
<=> ln(-3) = ln(3) + iπ

I don't know if this is formally correct because I don't know if that logarithmic identity holds for negative numbers, but it works.

[MathPhysicsFunwithGus]

Really love how you explain topics, math is so cool!!!

[ElevatorFan]

Log is used to find the exponent
If a=b^c, then log_b(a)=c

[carlosdelossantos]

And this video length is 3:14, some digits of pi

[reintsh]

At 3:02 "do complex numbers exist in the first place?" Well, do real numbers EXIST? Does any number exist at all? THINGS exist, numbers are abstract concepts. George Berkeley said: esse est percipi, exist is being observable. Numbers are truths that do not exist, yet they are truths.