/sci/ - Science & Math

[Anonymous No.4916149]

hey /sci/

can you prove the derivative of Y(x) = (-1)^x

is " i * (-1)^x * pi " ?

Genuinely not hmwk...

I was thinking about how wolfram would differentiate a function written as y*(-1)^x = 4 from one written as y = / (-1)^x... and then i realized why in the fudge is y= (-1)^x shown as a sinusoid... I thought that when X is even then y = 1, and when it's odd then y=-1... (i know, herp derp no even and odd in decimals, but that's part of the mystery... )

So when i looked at its derivative... i * (-1)^x * pi

I was even more puzzled, so the function is accelerating by "i*pi", hence the complex plane....

What am I missing here? There's clearly some sort of leap.. where can I read about this?

[Anonymous No.4916154]

accelerating by...i * (-1)^x * pi
the whole thing seemed circular, because you can sort of see the 2nd derivative would also been in the same family as the original one... and the 3rd too, and the 4th... etc....

this is so fascinating...

[Anonymous No.4916167]

Anyone math savy can tell me more about what i've stumbled upon?

[Anonymous No.4916180]

>OP and his thread

[Anonymous No.4916187]

cmon anyone im dying here to find out more

[Anonymous No.4916190]

bump

[Anonymous No.4916199]

:(

[Anonymous No.4916205]

Y(x) = (-1)^x
= e^[(ln -1)x]
Therefore:
Y'(x) = (ln -1) e^[(ln -1)x]
= (i*pi)*(-1)^x

You will learn how to get ln -1 = i*pi in complex analysis.

[Anonymous No.4916207]

OP:
-1=exp(i*pi)
For any complex number x, (-1)^x=exp(i*pi*x). The derivative is obvously i*pi*exp(i*pi*x) i.e. i*pi*(-1)^x (but you should rather write it as exp(i*pi*x) it's more appropriate). This function is actually not so different from x->exp(i*x) which basically turns around the circle of the complex plane with center 0 and radius 1. Your function is the same except it turns pi times faster.

[Anonymous No.4916214]

>>4916205
Your notation doesn't really makes sense. The complex exponential isn't injective and, as such, has no inverse function. The complex logarithm is not unique, a single complex number has an infinite logarithm just like it has an infinite number of arguments. Stating ln(x)=y only makes sense if y is real and >0. But the idea itself is not that wrong, though.

[Anonymous No.4916219]

>>4916214
I'll be nice to you and assume you are a very smart high school student. Take the compliment.

[Anonymous No.4916225]

>>4916219
Sorry, I'm a wise guy as well as former math student. :P

[Anonymous No.4916231]

>>4916214
infinite number of logarithms*
Sorry. By the way you can define the principal logarithm of a complex number (and in this case it makes sense but remember it is only one convenient logarithm among others): if x and y are real, then the principal logarithm of x+i*y is:

ln(sqrt(x^2+y^2))+2*i*(arctan(y/(x+sqrt(x^2+y^2))))

The other logarithms are generated by adding multiples of 2*pi to the arctan.

[Anonymous No.4916234]

I think you don't like my lack of mathematical rigor. I probably owe it to you to explain that my background is actually more in engineering. Perhaps that explains things better? *grin*

[Anonymous No.4916239]

>>4916231
Okay, give me a sec. I'm going to teach you a wonderful thing, as soon as I figure out what I wanna post.

[Anonymous No.4916243]

>>4916225
As said below, there is a single principal logarithm, but an infinity of logarithm in general for a same complex number.
I didn't mean to be despise you but ln(-1) always tend to sound weird to me. And I take your compliment, though high school is far behind now.
But let's forget about that. I have just finished a sophomore level cursus in math and physics (mainly). I'd be curious to know how far you got in mathematics, where you studied and if you had specialities of any kind.

[Anonymous No.4916245]

Let z = x + iy be a complex number. Then z can be written as:

z = A e^i*theta

where

A = sqrt(x^2+y^2)
and
theta = arctan(y/x).

^^ is a BIG HINT. I think you're smart enough to fill in the gaps...

[Anonymous No.4916247]

>>4916234
Well, no offense, but I was trained I France and they take those kind of details pretty seriously there. I was a bit ticked off because I could have got skilled by my teachers for that.

>>4916239

What do you want to teach me ? Something about mathematics ?

[Anonymous No.4916250]

>>4916247
No worries, completely understood.

And look at >>4916245 , if I've read you right, it should blow your mind.

[Anonymous No.4916252]

>>4916245
That's basically what I wrote above, except my formula (which is pretty horrible to read, I give you that) take in account the problem of periodicity. But I admit I'm glad to use the simplest way (module=sqrt(x^2+y^2), argument=arctan(y/x)) when treeating an engineering problem. I remember the other formule (the horrible one) because it was in my math lessons but I never really use it, I am ready to admit it.

[Anonymous No.4916253]

>>4916247
In the tradition of Descartes, et al, no doubt. /salute

[Anonymous No.4916254]

OP

question:

were the complex numbers introduced to solve this type of problem?

[Anonymous No.4916257]

>>4916252
It IS basically what you wrote above. I think you are missing some important conceptualization though: while algebraically the same, the form I have presented you is the representation of a complex number in polar form.

After you get another year under your belt, you will love it, very likely.

[Anonymous No.4916259]

>>4916254
Let me check for you. This is too important to get wrong. I know what I know about it, but I wanna get it right.

[Anonymous No.4916272]

>>4916259
>>4916254
Okay, in a nutshell: complex numbers have their origin, naturally, in the discovery (or invention if you will) of the number sqrt(-1) by mathematicians trying to solve polynomial equations with "no" solutions. The development of i = sqrt(-1) allowed for all n-th degree polynomials to have exactly n solutions (including repeats).

But it wasn't until electrical engineers adopted the use of i in their field that contemporary use really took off. The development of all electronics rely heavily on the use and manipulation of complex numbers.

[Anonymous No.4916279]

>>4916257
So this was it. Sorry I didn't where you wanted to lead me. I know polar form of course (which is what I used in my first post when writing -1=exp(i*pi)). The horrible formula I used proceed also from a polar form interpretation (it is basically another geometric interpretation of the argument).
But I agree with you the interpretation is really pretty (though it become somehow mundane after three years).

The funny thing is that it works also for C1 class functions. French call it "théorème de relèvement C1 sous form trigonometrique" which more or less means "trigonometric form of the C1 increasing".
Basically a C1 complex function from R to C that never becomes equal to zero on its definition interval I can be written as following:

f(t)=rho(t)*exp(i*theta(t))
where rho is unique and strictly positive and theta unique modulo 2*pi. You even have some interesting relations on the derivative of rho and theta.

[Anonymous No.4916285]

>>4916279
I think us electrical engineers love your invention more than you mathematicians do. You seem very ho-hum about it!

[Anonymous No.4916288]

>>4916272
To further your point: anything that deal with periodic signals has something to do with complex numbers. As you've seen above, complex numbers make up a plan (while real numbers make up a line) in which periodicity can be seen as rotation.
Since it happens that periodic signals are the most useful to transmit information, the technologies of transmission is essentially grounded (as for the mathematical part) on complex numbers.

[Anonymous No.4916292]

>>4916288
Indeed. You've made my day, sir.

[Anonymous No.4916294]

>>4916285
Let's say that engineer like when math enables them to fool around with awesome toy. Mathematicians only care to fool around with math. Complex analysis is a huge thing, still, and is useful almost everywhere in mathematics (in analysis as well as algebra) in physics (electronics but also thermodynamics and even hydrodynamics) and engineering.

[Anonymous No.4916298]

>>4916294
Be noted however, that I am (at least in the spectrum of engineers) one of the more "theory appreciative" ones.

In my second life I would study pure science/mathematics. If only...

[Anonymous No.4916300]

>>4916219
I think he was saying that -1 lies on a branch cut...

[Anonymous No.4916302]

>>4916300
Sorry, I've long forgotten the finer details of that topic...I spotted some minor errors in his statement and I assumed something that turned out to be, somewhat wrong, I suppose. So I apologize if there was offense taken.

But in my defense, I was joking. :3

[Anonymous No.4916308]

>>4916298
Depends on where you work but it's probably possible to work close to the theoretical field while being an engineer. For instance with the CERN or the guys who work on fusion in the US. Well actually I'd be interested to know more about your career and your studies. I am myself still a student and I'd like to have more infos about my possible future jobs.

[Anonymous No.4916313]

>>4916308
Yeah, was just going to answer your Q earlier. Bit about me:

-Berkeley undergrad
-UC Davis grad (withdrew)
-May attend grad again in the future
-Also exploring other career paths
-Currently not working at a paid job. I volunteer and sometimes attend a local college at the moment.
-Goals? Sky's the limit!

[Anonymous No.4916325]

>>4916313
I see...so you're still wandering a bit now ? Well I hope you can find a job soon, these times are pretty tough. Have you already worked in a science-related company (as an employe or even as an intern) ? I'd like to have an idea about the kind of science you do when you really start having a job. And also (more focused question) are the foreign student/graduates appreciated in the american science community ? Are the foreign school/college well recognized ?

[Anonymous No.4916337]

>>4916325
Oh, don't worry about me. I might have been cause for concern before, but I've recently had a personal renaissance, if you will.

I have interned at Intel before actually, but that was short lived, I quit. Mainly because it was not very science-oriented at all, I found out.

Regarding American attitudes towards foreign students/graduates: They occupy a large percentage of the graduate programs across the country, so no cause for concern there. And companies hire certainly hire them. Probably the most important thing is to be up on your English. If you communicate effectively, you will be fine.

Re: American science community. Pretty sure you will have no problems fitting in. See above comments about your chances of making it in industry; similar comments apply here too. Communicate well, and you should be fine.