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ArcTan really bothers me, so is there a way to make Mathematica write Logarithm instead of ArcTan? What I'm after is some kind of a ArcTan to Log converter such as the following list from Wiki

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3  
Try TrigToExp, e.g. TrigToExp @ ArcTan[x] works as you seem to expect –  Artes Jun 5 '13 at 16:05
    
That's awesome, thanks a lot man. You should post this as an answer. –  mapel Jun 5 '13 at 16:06
1  
So now I'm curious. What's bothering you in ArcTans? –  Sjoerd C. de Vries Jun 5 '13 at 18:03
    
@Vries They are way too restricted for me, $\log$s are much more general and easier to work with imho. –  mapel Jun 5 '13 at 19:32

1 Answer 1

up vote 5 down vote accepted

The function you need is TrigToExp, e.g.

TrigToExp @ ArcTan[x]
 1/2 I Log[1 - I x] - 1/2 I Log[1 + I x]

There is an inverse function for TrigToExp, namely ExpToTrig

ExpToTrig[ 1/2 I Log[1 - I x] - 1/2 I Log[1 + I x]]
ArcTan[x]

They both are Listable:

Attributes @ {ExpToTrig, TrigToExp}
{{Listable, Protected}, {Listable, Protected}}

Therefore you can use it e.g. this way:

TrigToExp [Sin[x] < x < Tan[x] ]
 1/2 I E^(-I x) - 1/2 I E^(I x) < x < (I (E^(-I x) - E^(I x)))/(E^(-I x) + E^(I x))
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