Understanding Notation for Hyperbolic Functions with Complex Numbers

[Gonzolo]

Hi, if you see:

cosh kx (1 + i),

do you consider the (1 + i) to be multiplying the cosh or inside the cosh ?

i.e.

cosh kx (1 + i) = (cosh (kx))*(1 + i)

or

cosh kx (1 + i) = cosh ((kx)*(1 + i))

I saw this in a thermoconductivity bible from the 50's written by a really smart guy, so there must be some standard to this notation. I suspect it is the latter, but need to be 100% sure.

 

[pnaj]

I'm afraid to say that it is ambiguous ... so maybe post up how the book got to this stage.

The thing is, if you've got to this stage of an derivation/explanation/proof or whatever and are not sure, then maybe you have missed something earlier ... it should be pretty clear from the context.

pnaj.

 

[Gonzolo]

The book skips many stages. It has a differential equation and then jumps to the solution f(x,y,...,phi) with phi = f(cosh...). Perhaps it is explained in an earlier chapter.

 

[uart]

Yeah I agree with pnaj, it's ambiguous notation. With this type of thing it's often clear from the context what it is supposed to mean, however I still find it kind of annoying.

It's funny that you posted this now, only today ago I was helping a student with a past exam question that read :
Prove the identity that cos (a+b)x + cos (a-b)x = 2 cos ax cos bx.

It's the same type of ambiguous notation. Of course if you know your trig expansions it's pretty easy to tell straight away that it must be the whole "(a+b)x" etc that is the argument of the cosine and not just the "(a+b)". Personally I'd still rather see another pair of brackets and have it in a form that even something as dumb as a computer could unambiguously understand. :)

 

[pnaj]

Gonzolo,

Post up the solution ... we should be able to infer from that.

pnaj.

 

[Gonzolo]

Here is the complete solution. It is pretty heavy.

 

[pnaj]

Sorry Gonzolo, but I can't seem to be able to open the doc ... I keep getting directed to a page that tells me I'm not logged in (when I clearly am).

I'll try again later.

 

[Gonzolo]

There we go (a similar problem, slighter simpler, had sinh instead of cosh), the sin()exp() are inside the summation :

 

[pnaj]

Ok ... that's pretty clear.

The writer must mean $\sinh (kx(1 + i))$, rather than $\sinh (kx)\times (1 + i)$, because otherwise, the $(1 + i)$ terms in the solution for $A$ would simply cancel.

pnaj.

 

[Gonzolo]

D'oh!