It does exist - but it's a multivalued function, so you need a branch cut.  See (halfway down the page):
http://mathworld.wolfram.com/InverseHyperbolicCosine.html

john

David Cournapeau wrote: 
Pierre SCHNIZER wrote:
  
Hello,

	I don't know if that only happens on my machine....
For numpy 1.0 b1 I get

<snip>
In [241]:numpy.arccosh(0)
Out[241]:nan
    
I get the same results, but this is logical... acosh is defined by 
0.5(exp(z) + exp(-z)) for z complex, and this cannot be smaller than 1 
for z real, and never equal to 0, so arccosh(0) does not really make 
sense to me,

cheers,

David
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