studiot

Then please look more carefully at the facts of what I actually said. Yes I offered some pleasant chitchat background about a possible evolutionary connection. But the main point I wanted to make was entirely contemporary. If you find it difficult to follow the logic try it the other way round. Work in terms of distance. Given that the distance from you ears to your brain is about 70mm, something travelling three times as fast will arrive at the same time from a distance of 210mm. It would be impossible for you to react to any change in audible signal that originated within this distance, within the transit time.

Are you really saying you can have an explosion without any explosive? Very confused.

I don't fully understand what you are trying to compare. There are two separate 'players' in an explosion. The active explosive material The passive target material. The progress of an explosion depends in part on the transfer of energy from the reactants to the target. So which part are you debating as affected by gravity?

Sorry you lost your typing. That happens to me all too often. Surely auditory sensors evolved pre hominid? Do we know for (reasonable certain) that the speed of our nerve signal transmission is significantly different from that in the first hominid (who came equipped with ears)? I suspect there is not enough difference to affect the simple estimate of time and distance I made. I don't see how faithful transmission of a signal over long distance would have been initially important?

Yes I think we are all agreed on this.

Good morning, function. Thinking about this further it may be more important to you to consider how you are going to use the data from the german paper. After all you are not going to repeat their survey or statistical analysis, just use the results. I have sent you some stuff on box plots which may be of interest. It is likely that the figures quoted (percentiles and median) were derived from a frequency plot (cumulative or otherwise). That could indeed lead to the median not coinciding with the centre of the age range. It would, however, be illogical to miss out ages because there is no data for that particular age so the axe axis must be complete.

Yes so long as you do not try to make injections from the reals to the rationals.

I did wonder at the OP's use of fractions since it implies his underlying set is the set of rational numbers, not the reals. Which is intended needs to be made clear before constructing a proof.

In which case there is more than you have stated since the age must be plotted as a cumulative frequency against some flu event. This techniqe is useful as it allows an open ended age range to be divided into quartiles. The median is the centre of the the central 50% (ie the inner two quartiles) of the frequencies of catching or recovering from or transmitting or whatever the flu. Edit You have a PM

Hello, function, please recheck your original. I don't see how the median can be 16, I think it should be 19.

That was just an example. I don't choose difficult ones for the sake of it. And you did ask about the connection to trigonometric functions. How did you get on with the Wiki article? The idea was to help you find the right search phrases and terms to look up. Just like for trigonometric functions there are some easy to calculate values and the connection to trig I mentioned allow these to be used. But yes there are tables of gamma function available, or if you could find some other way to calculate them, you could put random values into these formulae. For instance there is a table in the back of Kreysig 'Advanced Engineering Maths'. You will also find a useful connection to Laplace transforms here. I really am struggling to know what level to put an answer as you don't reply to my questions. which are only designed to help. I meant what I said about cooperation.

Do you understand the difference between an identity and an equality? The gamma function finds use in differential equations, including those such as Bessels, which do not have solutions in terms of elementary functions. Since another word for solving a differntial equation is integration, the gamma function can be used to simplify certain integrals. This is where its connection to trigonometric functions through the reflection formula ( http://en.wikipedia.org/wiki/Reflection_formula )comes in. For instance [math]\int\limits_0^{\frac{\pi }{2}} {\tan \theta d\theta = } \int\limits_0^{\frac{\pi }{2}} {{{\sin }^{\frac{1}{2}}}\theta {{\cos }^{\frac{{ - 1}}{2}}}\theta d\theta = \frac{1}{2}} \Gamma \left( {\frac{3}{4}} \right)\Gamma \left( {\frac{1}{4}} \right) = \frac{\pi }{{\sqrt 2 }}[/math] The issue of inverse functions is more tricky and brings in the subject of convergence quite strongly. You need to understand the idea of domain and co domain (or range) for functions and inverse functions and its implications which leads to the idea of 'radius of convergence' in complex analysis. This has its counterpart in the 'interval of convergence' in real analysis.

+1

I see that there have been quite a few views since I posted so here is some background. Engineers use tables/charts of the Joule Thompson coefficient, which is basically the ratio of delta T to delta P Actually it is the slope of the constant enthalpy line on a T_P diagram. http://en.wikipedia.org/wiki/Joule%E2%80%93Thomson_effect This reference has a value of 0.055 oK /bar for this coefficient at 300oK The question basically drops from 108 to 105 Pa ie (1000 - 1) x105 Pa or 1000 bar, Which implies a temperature rise of 55oK Here is someone reporting burning himself on the temperature rise from 200 bar cylinder of Helium throttling to 1bar. https://www.physicsforums.com/threads/i-got-burned-by-a-helium-tank-valve-how.627972/ An typo crept into the previous post b should be 2.3 x 10-5

This repeated violation of forum rules that you signed up to is counterproductive. Your first post is a mixed up jumble that even normally mild ajb comments on I don't know why you are determined to restrict the discussion to Eulers formulae or why you specified this had to be done in your first post. I don't know where you are coming from that you need to solve complex analysis problems, but this is particularly common in electrical engineering. Eulers formulae forms the basis of some useful analysis methods, but it is only part of the story and not used directly in much real worl working. You need more than this, in particular some knowledge of complex analysis to understand the working rather than just take the formulae on trust. I will work through a simpler problem than you have presented in post#1; the problem is simpler but embodies most of what ( I think/guess) you need. Evaluate sin-1(2) This does not converge for any real number, but does converge for some complex number, z = (x+iy); where x and y are real. We proceed as follows 2 = sin(x+iy) = sin(x)cos(iy) + cos(x)sin(jy) = sin(x) cosh(y) + icos(x)sinh(y) Equat real and imaginary parts sin(x)cosh(y) = 2 cos(x)sinh(y) = 0 From the second equality either cos(x)=0 or sinh(y) = 0 If sin(y) = 0 then y = 0 and cosh(y) = 1 This makes the first equality impossible since x is real Hence cos(x) = 0 Hence [math]\sin \left( x \right) = \pm 1[/math] Hence from the first equality [math]\cosh \left( y \right) = \pm 2[/math] But since y is real [math]\cosh \left( y \right) \ge 1[/math] Hence cosh(y) = 2 and sin(x) = 1 Hence [math]x = \frac{\pi }{2} + 2n\pi [/math] where n is any integer and [math]y = {\cosh ^{ - 1}}\left( 2 \right) = \ln \left( {2 \pm \sqrt 3 } \right) = \pm \ln \left( {2 + \sqrt 3 } \right)[/math] Hence [math]{\sin ^{ - 1}}\left( 2 \right) = \frac{\pi }{2} + 2n\pi \pm i\ln \left( {2 + \sqrt 3 } \right) = 1.57 \pm 1.32i + 2n\pi [/math] Which is complex. Other examples might be Using [math]{e^z} = {e^{\left( {x + iy} \right)}} = {e^x} \bullet {e^{iy}} = {e^x}\left( {\cos y + i\sin y} \right)[/math] [math]e\left( {2 + 3i} \right) = {e^2}\left( {\cos 3 + i\sin 3} \right) = \left( { - 7.32 + 1.04i} \right)[/math] There are many more, but we cannot go over them, unless you stop attacking me and start cooperating.

A word of warning to anyone trying this. Ethanol cannot be directly distilled to 100% from water. The so called 'absolute alcohol' you find in the lab has has some other substances (poisons) added to perform the final separation. These substances are later removed as far as practicable, but traces remain. Every now and again we read of someone using a bottle of lab alcohol to beef up a punch and the guests falling ill, even dying. Do not play with lab ethanol.

Our ears did not evolve to listen to continuous sounds such as music. They evolved to react to change. For example the sudden rush of sound caused by a predator. The important factor in this is the time the change takes to reach its destination. Now in the time it takes for a predator to travel say 10 metres towards you, according to your speed figures which I have no reason to doubt, a nerve signal can travel about 3 metres. But the distance from your ears to your brain centre is only about 0.07 metres. So there is ample time for a change signal to arrive.

Did someone in Russia murmer something about a troublesome priest? In the future how will students of history compare the outcomes of the two events?

Your words, not mine nor imatfaals or anyone elses. In response I stated that and further gave link to a respectable webpage on the subject referred. You responded by claiming that I am lying, and irrational Categorically , your post#1 does not contain any reference to the hyperbolic sine or cosine, So who is lying? Finally you go some way to understanding what I am saying So if x is imaginary then ix must be real. And if x is imaginary why would anyone bother to add the i, unless they wanted the result to be real? You cannot have your cake and eat it.

So far you have not addressed a single one of my questions about your lack of rigour, each time avoiding the direct question with a personal attack or self contradiction. I asked a perfectly reasonable question if x is imaginary what is ix? Instead of answering you state (wrongly) that I have not read you first post. You further claim material that was not in it and casitgate me for not finding such non existent material.

That's twice now you have substituted personal attack instead of mathematical argument. That apart, you have now contradicted yourself I cannot see any reference to the hyperbolic sine or cosine in your first post, perhaps you misunderstand Euler's Identity? http://en.wikipedia.org/wiki/Euler's_identity Your first post did, however, make very clear that you regard x as a real number, as does Wikipedia.

So it is a function, and roughly is not exactly. I have already agreed I could have said that the attenuation is a function of both material, thickness and the radiation concerned and that would have been better.

If you know the answer why ask the question? If x is imaginary, is ix real or imaginary? If you wish to work with a complex number you should use z, but should if you wish to use x instead (and that is being obtuse) then you should not also use ix.

There are no solutions as you have chosen to write it. I copy and pasted your equation into Wolfram and it timed out without solution. Please state the question properly.

I didn't say the relationship was exponential. But it is definitely untrue to say that charged radiation is not attenuated by the interposition of matter its the path, or that the reduction in measured radiation is not a function of the thickness of the matter impeded path. Agreed my wording was not perfect, but yours was no better.