Maths question on proving arctanh

[MumboJumbo]

Hey well trying to revise and ive managed to get my self into a mind block. Its a quick question for anyone who knows how to do it.

Im trying to prove arctanh and this is what i have so far but im lost at filling in the gap . as you should get

x(e^(2y) + 1) = e^(2y) - 1

e^(2y) = x(e^(2y)+ 1) - 1

So any help would be appreciated, its probably quite simple and im being dumb. Cheers.

ignore "<><>" thats just me trying to space it out as you can't do it on these forums.

-------------------------------------------

y = argtanh(x)

<><> e^(y) - e^(-y)
x = -------------------
<><> e^(y) + e^(-y)

<><> e^(2y)- 1
x = ----------------------
<><> e^(2y)+ 1

<Unsure of this point>

<><><> 1 + x
e^(2y) = ---------
<><><> 1 - x

<><><><> 1 + x
y = (1/2) ln ------
<><><><> 1 - x

 

[frying_pan_cat]

I'm slightly unsure as to what you mean when you say you are trying to "prove arctanh" but going from what you've got there

from:
<><> e^(2y)- 1
x = ----------------------
<><> e^(2y)+ 1

multiply both sides by
e^(2y)+ 1
giving you the line you had at the start:
x(e^(2y) + 1) = e^(2y) - 1
then your next line was wrong as there should be a plus there giving:
e^(2y) = x(e^(2y)+ 1) + 1
obtained by adding 1 to both sides of the previous line. You can then multiple out the bracket and collect the e^(2y) terms on the left giving you
e^(2y) (1-x) = x + 1
then divide through by 1 - x and you have the next line you've got

Hope this helps

 

[Grimbough]

xe^(2y) + x = e^(2y) - 1

(x-1)e^(2y) = -(x+1)

e^(2y) = -(x + 1) / (x - 1) = (1 + x) / (1 - x)

2y = ln | (1 + x) / (1 - x) |

y = 1/2 ln | (1 + x) / (1 - x) |

 

[MumboJumbo]

annoyingly i just did it my self and came back to tell you guys. But you did it anyway. Cheers . Dunno how it took me that long tbh. Need more coffee!

 

[chrism]

god my head hurts!

 

[SlyReaper]

How do you prove arctanh? You prove theorems, propositions, lemmas and corollaries. Proving a function makes no sense.