>>I am researching the usage and distinction of angular
>>measure from linear measure and how it impacts the complex
>>exponential functions, especially for use in dimensional
>>analysis in applied maths.
>
>Quite honestly the problem may be beyond my comprehension.
>In any particular case, I (if at all) use the units that I
>find more convenient.
>
>>I am interested in locating any references that clarify the
>>use of Angle as a distinct measure.
>
>wikipedia?

I have been looking there. In particular the hyperbolic page https://en.wikipedia.org/wiki/Hyperbolic_function with its swept area definition

This is supported by https://en.wikipedia.org/wiki/Inverse_hyperbolic_function (ref 1)

(not that I believe that approach. It is more of a Kepplerian law of planetary motion)

I posted the question here because of your references...

>
>>In pure maths, the distinction between linear measure and
>>angular measure is regularly omitted, leading to bad
>>calculations. An example would be the dimensional assumption
>>that Torque = Energy. Both are Force x distance.
>
>Well. This is rather superficial. Torque is a vector, Energy
>a scalar.

On a 2d plane, the 'vector' is always zero, as it is 'out of the page', hence the common mistake in general engineering dimensional analysis of ending up with N.m = m.N (metre.Newtons) for both Torque and Energy, when in fact, the torque must rotate through and angle to transfer energy.

>
>>In fact Torque is Energy/Angle.
>
>At best, this is true for the size of Torque. Once you shed
>the physical meaning of the items involved you can probably
>arrive at other apparently meaningless assertions.
>
>>This is the typical lever arm.
>
>You mean "lever arm's dimension," right?

I was just advising of the domain of use.

>
>>The regular notation is then r.exp(i.theta), separating the
>>real radial part, which has dimensions of perhaps Volts (on
>>a linear measure), and the imaginary exponential phase
>>(angular) part.
>
>>If however exp(rho+i.theta) was used, we would be confronted
>>with the conundrum of how to handle the two possibly
>>differing units of measure.
>
>It may be up to you to make the units compatible. It does
>not make a lot of sense adding apples and oranges, but both
>are fruits.

Exactly. There may be instances where the 'unit & dimensions' of the real and imaginary parts could be different. The specific case being if angle is considered as a different type of dimension.

>
>I'd think that unless you have some units compatibility you
>should not use exp(rho+i.theta).
>
>On the other hand, exp(rho)×exp(i.theta) should help
>one escape confusion.

This is my "solution" for the case where we have machine based checking. It allows a rule to be created for separating the angle part from the radial part within the otherwise complex number. They become either real or imaginary.

>
>>Complex numbers already require
>>the two parts to be kept separate which could offer the
>>chance to attach differeing units to each part when using
>>that transformation. Alternatively the rule could be to
>>avoid complex numbers where both parts are non zero (as
>>above).
>
>Again, I am at a loss to grasp the idea of avoiding complex
>numbers. The are handy and useful and have been around
>fruitfully for a while now. I believe that, whenever
>necessarily, in order to avoid confusion, people often apply
>common sense. May it be the case with complex numbers and
>their exponents?

There never seems to be enough common sense to go round...

I'm not trying to avoid complex numbers, but rather trying to understand how the pure maths usage can create errors in applied maths calculations, particularly in the case of tools that support unit (dimensions) checking. I have see users add a 2.pi here and there to make the numbers 'right' when in fact they had got their formula dimensionally wrong.

At the beginning, my question was really how should one handle the sine of a complex value. Normally we are being asked for the sine of a real angle.

If we are asked for the sine of an imaginary value we convert it to the sinh function of the value, but made real <(-i).sin(i.x)=sinh(x)>. But we don't call it an angle, or do we?

Philip

>At the beginning, my question was really how should one
>handle the sine of a complex value. Normally we are being
>asked for the sine of a real angle. >simple relation to the exponential when complex numbers are
>invoked]

Sine and cosine appear in contexts other than involving rotations. Waves exist everywhere. So, I think, interpreting their argument always as an angle may lead to confusion, let alone being hard to justify. In the complex domain, I think of sine and cosine as functions from one region to another thus, while excepting possibly different ranges for the real and imaginary parts of the argument, I still think of them as having interchangeable roles, at least dimensionally. exp(z) maps a strip to a plane (less a point). The strip is usually depicted being horizontal. But you can rotate it, say by 45°, by mixing x & y so that the result is a strip with a symmetric occurance of x and y - and this comes out naturally suggesting that it would have been artificial to call the original y an angle and the original x a distance.

>If we are asked for the sine of an imaginary value we
>convert it to the sinh function of the value, but made real
><(-i).sin(i.x)=sinh(x)>. But we don't call it an angle, or
>do we?

No, I believe, we do not. And, as above, we do not always do that even in the real domain. In the wave equation, the relevant parameter is actually distance, not angle.