Question

Does this function have a specific name?

Answer 1

\[\LARGE f(x)= e^{|\ln x|}\]

So for any rational number greater than zero it will always give you a sort of "magnitude" away from 1 where numbers greater than 1 are like positive numbers and numbers between 0 and 1 are like negative numbers.

For instance, \[f(\frac{5}{2}) = f(\frac{2}{5})=\frac{5}{2}\]

Answer 2

thats an interesting function

Answer 3

e^ | ln x | = x for x >= 1

the interesting part is what occurs on (0,1)

Answer 4

for x inside the interval (0,1)
we have
e^ |ln x | = 1/x

Answer 5

this formula also applies for nonrationals
you can use as as example
x = 1/e

Answer 6

by the way, i think we should stop calling numbers irrational

Answer 7

Yeah it sort of is the "length" if we consider 1 to 0 to be equivalent to 1 to +infinity.

I know in the past I've sort of considered this, but I don't remember why. It's also related to 1/x, which is a hyperbola.

I think you're right, about not calling numbers irrational. It's sort of like getting interesting if you think in terms of this: \[\LARGE y= e^{(e^{i2\pi/n})x}\] See, this function is its own nth derivative. ;P

Answer 8

yes thats true

Answer 9

because of all those exponent properties

Answer 10

But we can take it deeper, and it gets interesting. I don't know, I'm just sort of playing around here, but I really like this function I just don't know how to use the "absolute value of the exponent" for anything currently.

Answer 11

( i havent done many derivatives on complex numbers, but that makes sense intuitively)

Answer 12

just using the chain rule and basic complex number definitions

Answer 13

Yeah it's definitely fun and makes sense. This is what I mean about going deeper.\[\huge e^{e^{e^{i2\pi/n}}}=?\]

Separately, one reason why I'm interested in finding a sort of "magnitude" away from 0 is that it sort of allows us to map all the numbers greater than 1 to a simple finite interval. Maybe this allows us to manipulate things and play with infinity a little?

It's not quite clear to me what complex exponents are either, since positive and negative have definite meanings. Curiously, i^i is a real number for example.

Answer 14

in the nth derivative you have

e^( i 2pi /n ) * ... e^(2pi/n) * exp ( exp( i 2pi/n) * x

Answer 15

and e^(2pi i / n * n ) = e^(2pi * I ) = 1

Answer 16

also the function above has the property

f(x) = f (1/x)

Answer 17

I originally discovered this sort of ability to make a function its own nth derivative when I was playing around with the definitions of sinx, cosx, sinhx, and coshx along with plain e^x and noticing how they had this odd cyclic derivative structure, so I wanted to generalize it and this is how I did it in case you were curious.

Answer 18

yes

Answer 19

i saw you typeing and you stopped

Answer 20

Any ideas about using 1 as the center of a number line with 0 and infinity as end points and no negative numbers allowed?

Answer 21

hmmm

Answer 22

you could shrink (-oo,1) into (0,1) , but thats not putting 1 at the exact center

Answer 23

but i could be thinking of something else

Answer 24

We have the regular derivative for n>1 but what is the derivative of the same function 'flopped' over and squeezed between 0 to 1? It sort of goes from x increasing at a regular interval to now increasing at a logarithmic interval. So is sort of is like our interval is a function itself and no longer is d/dx(x)=1 kind of thing going on if that makes sense?

I guess we could do that @perl but I don't know if that would get us anything or not. It's essentially the same sort of transformation is all I'm saying.

Answer 25

is there a nice formula for the nth derivative of

e^ (e^ [e^( i 2pi / n) * x ] )

Answer 26

oh e^ e^ | ln x | is e^x for x >= 1

Answer 27

Not that I have any idea of I'm afraid, I just don't know lol. I don't even know what the significance is, but it is interesting to me that i has a sort of recursive definition.

Answer 28

yeah its cool

Answer 29

when you start to do math for a while, you start collecting 'specimens' that are interesting, as a zoologist might for a rare butterfly
unfortunately maths people dont have a special word for it
we call them 'examples' :/

Answer 30

interesting examples / function

Answer 31

functions*

Answer 32

is it true that

f(x) = x^2 maps (0,1) to (1, oo)
and
f(x) = ln x maps (0,1) to (-oo, 1 )

Answer 33

Yeah it's nice to have interesting things in your mind to play with from time to time, some things just sort of work themselves out over time and develop into new things. It's pretty fun haha.

I think my original motivation for playing around with this is that I was convinced that the function y=x^2 actually turned around as you approached infinity and created a sort of "sine wave" that was stretched out logarithmically far. I don't remember why I thought that though but I will play around and see if I can't draw it out.

Answer 34

Here was what I had suspected, it was x^3. Make sure you see that this is this graph rotated and squished logarithmically after 1 to infinity