How much energy it takes to create an universe in which the expansion goes on forever and accelerates forever?

The reason why I ask this, is that it simply doesn't seem to make sense. Let's say that the universe indeed is expanding and that the expansion is increasing in speed. How much energy was required to not only create the universe but also increase the speed of accelaration?

Unless something outside the universe is constantly feeding the universe with energy so that the universe can expand, then I don't see how expansion can be possible. Because currently, it is believed that the universe will expand FOREVER at increasingly speeds. It would take an infinite amount of energy to do that. It just doesn't make sense.....

exactly....

if the universe was allowed to manifest energy as expansion... then why would anything have ever formed within space?????

all the energy of creation would go to expansion first... and never bother to form matter..... why would it...

forming matter is the opposite to expanding space...

i dont believe space is expanding... its just moving 4 dimensionally, and so looks like it is........... looks like.. thats all.

-MT

Moving where?

If galaxies are moving in relation to one another, then that signifies expansion. Unless there's a pattern of movement. If they are moving all together in the same speed and towards the same direction, then, and only then, you can see it as the universe as a whole moving towards somewhere.

My answer is that there's a pattern of movement. Read this thread (http://www.sciforums.com/showthread.php?t=51297) and then put many drains in place of the cup (and in a huge swimming pool). Then watch the pattern of movement of the water. What do you see? ;)

yabadaba

Looks like I have a fan......:D

How much energy it takes to create an universe in which the expansion goes on forever and accelerates forever?
...
Unless something outside the universe is constantly feeding the universe with energy so that the universe can expand, then I don't see how expansion can be possible. Because currently, it is believed that the universe will expand FOREVER at increasingly speeds. It would take an infinite amount of energy to do that. It just doesn't make sense.....

I hope I am not wasting my time; I.e. that you will try to follow, what may be a little bit strange type of argument for you, or thought pattern. I make the effort as you have asked a good question that reflects some thought. I hope you will make the effort to follow me. Deal?

You (I think) wonder how could the universe expand for ever, if gravity is always attracting everything towards each other thing. Correct? (Anyway that is the question I will answer. - I only briefly comment at end on the acceleration of the expansion, but show now with simple math, that it does not reqire infinite energy to expand "forever.")

To answer this question, we need to know if the energy required to expand forever is finite or infinite. Lets build a simple model to try to find out. I am assuming that you (or some reader also interested) do not know calculus, which would make the answer relative easy to get, so I will approach the question sort of the way calculus was discovered or invented - by a lot of small step calculations, which we will add up later. Also to keep is simple, I will only calculate the energy required to separate from each other only two spherical bodies, one of mass “M” and the other of mass “m”, not expand the whole universe. (It is really the same problem - just imagine each has half the mass of the universe, if that helps you believe this. - more later also.)

Lets begin when the two masses have a separation of 1. (one light year, one mile, one Km, one what ever you like) but the sum of their radii is less than 1 so there is a space between them. Thus, the gravity force initially between them is G M m / 1x1, but I will chose the units of mass (7 solar masses, 15Kg, N bathtubs of water, or what ever is necessary so that G M m = 1) Thus initially there is one unit of force between them. I do not know, or care, what is the magnitude of this force is if expressed in common units, like pounds, etc.

Now lets think about how much work (energy) is required to pull them apart a little, say until their separation is 1.1:

Well work is the product of force acting times the distance moved. That distance was 0.1 and the force initially was 1 but it decreases as they separate and this makes it a calculus problem to do exactly, so lets assume that we are willing to do a little more work (or have a little more, but still very finite energy available). That is, we know that the energy required to do the work of pulling them apart by 0.1 length units is less than 1 x(o.1) or in words the initial force times the entire step taken. That is to separate from 1.0 to 1.1, or 10% of the starting separation will not require more than 0.1 energy units (actually a little less).

Now lets take a second 10% step apart, but we will notice that at the start of this second step the force of attraction between the two bodies is only 1/(1.1x1.1) or 1/1.21 and again we neglect the fact that the force is decreasing as we now move out 10% from 1.1 to 1.21 or we once again find that the energy required for this second step is less than (1/1.21)x (1.21-1.10) = 0.11/1.21 = 1/11, which is less than to make the first step even though the second step was bigger and which I can call the “ample energy of step 2.”

I will let you do he next step, if you like, to confirm that the third step’s “ample Energy” (more than enough) is even less than the “ample energy” of the second step. (later I will give a math proof this is true for every step, when compared to the immediately previous step (and certainly to any earlier one).

Clearly I can get to infinity if I take an infinite number of steps, each at least as big as the previous one (and every one of these 10% step is actually bigger than the previous one.) The question is thus reduced to:

“Is the sum of an infinite number of these ever decreasing "ample energies" infinite or finite?” Answer is, as I will soon show, that it is finite. Consequently, it only takes a finite amount of energy to separate half of the universe from the other half by an infinite distance. (Obviously, now that I have each half infinitely far from the other, each could be split into two parts which are then infinitely separated from each other etc. again, and again etc until I have every atom separate from every other by an infinite distance. I only need do his a finite number of times, if the universe is finite. Because a sum of a finite number of finite quantities of energy, is finite, is why I earlier said “It is really the same problem as just separating the two halves of the universe.” That is if the universe is finite and if the energy to separate it into two halves is finite, as I will soon show it is, then energy required to separate every atom in it from every other by an infinite distance is also finite, very big but finite.)

I suggest you do the next or third step, but I will tell some tricks to avoid this work. (I am old and lazy, so I need to be clever to avoid working too hard.) If you are young or less lazy, I strongly suggest you do at least step 3, so you can numerically check to see that my tricks are valid. -Post the calculations for step 3, and if you need help understanding my “tricks” I will try to explain them more fully. (If I fail to return here, and you need help, send PM to me.)

Tricks:
(1) First, I am going to sometimes call step 2 “step A” and sometimes call step 3, “step B.”
(2) Next I buy or make a new, longer, ruler to measure distance with. With it 1.1units on the old original ruler are exactly 1.0 new units. Thus, the starting separation of step A is 1.0 of the new ruler’s length units and the starting position for step 3 (also known as step B) is exactly 1.1 units on the new ruler (10% more, as before). That is, whatever was true with regard to any relationships calculated above between step 1 and step 2, is also true between step A and step B.
For example, the ratio of the “ample energies” in steps 1 & 2 was 0.1 to 1/11 or 1 to 10/11. Let me define “a” as 10/11. Thus, since I already know the ample energy of step 2 was 1/11 the ample energy for step B (also known as step 3) is 1a/11 or a/11, but I am not really interested in this, but do want you to note that step 4’s ample energy is just (a^2)/11 that of step 5 is (a^3)/11 etc.
(3) Now just as I found it convenient to change my ruler units, I am changing my energy units. So the first step energy in these new energy units is 11/10 of these new, much smaller (eleven times smaller) energy units. The second step's "ample energy" is 11a/10, but when you remember that a = 10/11 you realize that in these new unit’s the second step's “ample energy” is just “a” and also, I hope, notice that the ample energy for step 3 is just a^2 in the new smaller energy units. etc.
(4) Now lets make a paired list of step numbers and associated “ample energies” or AE:

Step # = 1.….…2...…3..….4.……….n……..
AE = 11/10.…..a…..a^2.…a^3.…..a^(n-1)….

(5) Now we know that the sum of these distance steps diverges (just a fancy term for saying no mater how big a length you want to imagine, a finite number of these every increasing steps will exceed that length, and if we like we could, in our imagination, take an infinite number of these ever bigger steps, we go to infinity. I.e. the infinitely long sum of these steps is infinitely large.)
(6) Our important question was “Is the total energy required to get the two halves of the universe infinitely apart finite or infinite?” That is, Is the sum of the second row of table above finite or infinite, even if we never stop adding additional terms to the sum? Well, as I told you earlier, it is finite (because a < 1.)
(7). If you do not already recognize this infinitely long series in the second row, it is called the “geometric series” and its sum is easy to calculate (and even easy to derive the formula for the sum) Google for more information or open most any simple math book that discusses simple infinite series. (You must do some work too. - My reward, is the hope that at least one reader learned something.)

SUMMARY: Without ever calling on “dark energy” ** to push the universe apart and without any use of calculus, we now know that only a finite amount of energy, for example kinetic energy, in the universe can separate all its parts infinitely far from each other. Gosh, after all that, if you have read all the way to here, I sure hope that was indeed at least one of your questions. :)
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**Dark energy is required to explain why the expansion of the universe seems to be at an increasing rate, but not just for it to expand "forever."

Hi Billy,

Thanks for taking your time, that is really good...

Let me summarize my answer:
1) Depends on how you define "infinity". It works well when you deal with infinitesimals, but not so well with large numbers.

2) You could argue that inertia keeps the universe expanding, and that would be a good argument except... that the universe is supposedly expanding so we can rule out any kind of inertia.

3) You didn't quite take into account the acceleration of expansion. When you write down all the numbers, you need to take into account the relationship between all the forces involved.

4) See my mortgage example

You (I think) wonder how could the universe expand for ever, if gravity is always attracting everything towards each other thing. Correct?
No. Even without gravity, it takes energy to "create" something.
That's why the thread is named "For How Long Can You Blow Up a Balloon?". A ballon has no force necessarily deflating it. When it is resting, there is no force at all. When you blow it, you can stop blowing it and tight it, so that it does not deflate. The universe can be seen in a similar way. The question arises when you have to do that forever, at increasing speeds and without blowing up the ballon (well, I'm not sure if the universe can blow up, but you get the idea... :D).

(Anyway that is the question I will answer. - I only briefly comment at end on the acceleration of the expansion, but show now with simple math, that it does not reqire infinite energy to expand "forever.")
Thank you. That was a good explanation, but unfortunately, it is not my question.

I am assuming that you (or some reader also interested) do not know calculus, which would make the answer relative easy to get, so I will approach the question sort of the way calculus was discovered or invented - by a lot of small step calculations, which we will add up later.
Yes, the answer is easier with calculus (go ahead, use it with me from now on).

“Is the sum of an infinite number of these ever decreasing "ample energies" infinite or finite?” Answer is, as I will soon show, that it is finite.
Not so fast. What is your definition of "infinite"?
You see, calculus works through approximation. There's never an ACCURATE answer in calculus. It takes you to answers such as "6.999999999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999999999 9"... which you define it as "7" to simplify it. Unfortunately, the same "infinite" concept does not necessarily work the other way around when it is defined as an unreachable number. This is my definition.

Here's why. Some people define "infinite" as a really big number. The problem with that is that this "infinite" is actually just a very big finite number, or a very small finite number. That's not infinite, that is finite. If you want infinite, then it will never stop either way. Calculus never stop either way. Calculus work through approximation. You use calculus to reach the above number, which results in the answer "7". It is a good enough approximation. When it comes to an extremely big number, such an "inifnite" number cannot necessarily be defined.

Yes, I know. Most people don't accept this, but that's what the number shows. Even in that number above, the "9s" go on forever. So technically is not 7, but it can be defined by 7. It works when you work with infinitessimals. However, when you work with trillions upon trillions upon trillions in a never ending chain of 0s, then that's a different story. And, believe me, it never adds up because it never ends.

So the calculus stuff- that's great. It works perfectly well with infinitessimals. But when it comes to huge numbers, it simply cannot define infinity.

So the simple answer is:

Depends on how you define "infinity".


3) Aaaaand... even accepting your mathematical model as true to this case, as the universe is increasing at accelerated rates, chances are that the energy required to expand the universe is also increasing. Therefore, for every decrease in the energy required you must also take into account the increase in energy required to accelerate the expansion of the universe. When you take into account all the variables, you see that even though the energy required may decrease as the space-time increases, the acceleration also requires an additional energy, which actually increases (and you did not take that energy into account).

Here are some possibilities:
A. The universe has a stable size. In this case, the following conditions must apply:
1- Energy required to expand the universe is equal to the opposing energy or it decreases with time
2- In the case of an accelerating universe, the energy required to expand the universe must increase in the same amount of the opposing energy
3- Obviously, there must be an opposing energy
4- The key words here are "balance" and "equal"

B. The universe expands:
1- Energy required to expand is stable or decreases while the opposing energy, if there is one, would have to be smaller in all cases
2- In the case of accelaration of expansion, the energy required to expand the universe must be greater then the opposing energy, or it simply needs to increase
.
.
.
You can really map out any scenario when you know all the data and you take all variables into account. In order for the energy of expansion to decrease, there must not be an energy of accelaration. If the universe is accelerating, then how come the energy required is actually smaller?


4) Let's take a look at your mortgage, assuming you have one. You must think I'm crazy, but in fact, the math is the same. Just the situation is different.

How do you pay off a mortagage? Basically, the principal payments increase while the interest payments decrease until you pay your balance. If you have a $100,000 mortgage and you pay 10% of interest per month, then in the first mont, you will pay mostly the interest and almost nothing will go towards the principle. This way, it maximizes the term and the profit from the mortgage because the amount of principal is what defines the amount of interest which you pay. Therefore, a long term mortgage will decrease exponentially as you go along.

(You can come up with numbers and a table if you would like)

Now flip the graph. Tada! You now have the same for the universe. As you can see, the acceleration actually increases very slowly, to infinity (not to 0 as the mortgage).

What you seem to be trying is to pay an infinite mortgage. You have an infinite principle and you are trying to pay the interest as well. Would you ever try to do that? ;)

depends on how you define infinity?
infinity no matter what you are talking about is endless
example:
death is infinity

edit
when you talk about numbers there is no number that defines infinity

Yes. Unfortunately most people don't think like that. :bugeye:

if space as a whole... was spinning on two axis's...
and if those axis's were at least 90 degrees apart...

then space would be moving in two directions at ones... without going diagnolly..

so.. without the need for expansion.... it would be moving 4 dimensionaly.

and so... looks like its expanding.. but isnt.

-MT

How much energy was required to not only create the universe but also increase the speed of accelaration?

Infinite energy. That energy is the magnetic energy. You know, when you have a goal, you have lots of energy. The universe also has a goal, that's why there's so much energy. That goal is unification. The closer the goal you are, the faster you go, just like with "gravity".

Take two magnets and put them close to each other: no energy is needed for them to come together, to unify, because they're only trying to restore the natural state of rest between them.

Magnetic energy is the only energy in the universe. No other energy is needed to explain it. The cause of everything is the goal (nothingness) which is already attained in the presence since it has no duration and hence no "material" existence.

Unless something outside the universe is constantly feeding the universe with energy so that the universe can expand, then I don't see how expansion can be possible.

Good thinking. The universe is created, made visible, in the presence. There can never be any visible causes, because every visible effect needs a cause. You can liken it to a TV broadcast, as long as the signal continues, the image on the screen will also exist.

But all this was already known over 5,000 years ago...

Depends on how you define "infinity".

Do you know how mathematicians work with infinity in real analysis (which proves properties of limits used in calculus)? For example, if f(x) = 1/x you can show that f approaches 0 as x increases infinitely. The 'number' infinity is never used since it isn't definable, but the concept 'as x increases infinitely' is easy to describe.

Saying lim (x->inf) f(x) = 0 is rewritten as
for any ε > 0, there is an N such that, if x > N then |f(x) - 0| = |1/x| < ε
if you look at positive x's, |1/x| < ε == x > 1/ε.

Note that ε > 0 - you're not saying that 1/x ever reaches 0. You're just saying that for any value you want - 0.1, 0.00001, 10^-1000, you can go far enough right on the real line (i.e. 'towards positive infinity') that f(x) will be that close to 0. Arbitrarily close. Thus we say that the 'limit at infinity' is 0.

To truthseeker:

You have some errors inyour concepts about calculus. It is not and approximation. For example the intergral of x is exacly (x^2)/2, not approximately this. In this general form there is no numerical value. If instead you want o speak of the "definite intergal" of x between 1 and 2, there is a numerical value: (4 -1)/2 = 3/2, exactly or 1.5 exactly, not "approximately."

Now some of calculus resullts (not calculus itrself) can be impossible to write in numerical form such as 1.5 is in numerical form. For example, the square root of 2 could be the result of some calculus computation. You know that any attempt to write it in numerical form, such as 1.414213....... , is only an approximation, but this is not "calculus's fault." - If a calculus calculation of this type produces the square root of 2, the it produced exaclty the correct answer.

Unfortunately, some calculus integrations are impossilbe to do (I think that may be true, but many that are certainly impossible for me, which may yield up their answers to some one else. Perhaps all might yield to some one smarter, more clever than any human. I.e. I am not enough of a mathematician to know if it has been proven that some well defined (analytic) functions can not be analytically integrated or not. Often complex integrands can not in practice be analytically integrated, so then if you still want an answer, you must result to "numerical integration" which is onlly approximate.

I had never though about it before now, but I have no way to judge the truth (or falseness) of the following simple hypothesis:

At least one analytic function exists, which can not be analytically integrated.

I am sure there are an infinite* number that no one currently living can integrate analytically.

There are several good mathematicians active here, Shmoe to name one, I remember because we have exchanged "one-way posts" (I.e. he can, and has, taught me, but there is nothing in math I could teach him.)

On the question of what, how and why etc. about "Dark Energy," I know much less than my very limited knowledge of math. I only know that I know nothing and lack the education to even learn anything about it, other than how to babble, with considerable misunderstanding, about it from distorted popularized accounts relating to some very complex math. I hardly can claim to really understand the observations that suggest the need for it. That is why I only briefly mentioned it in my first response to you.
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*I recognize two definitions for infinity. (1) A value greater than any specific value. (2) The limit of many function as the variable approaches some value, such as variable x approaches 3 in the function y = 1/(x-3).

It is quite possible that neither is considered adequate by real mathematicians. I am not sure, but strongly suspect there are subtitle differences between these two definitions.

I think this because definition (1) clearly relates to only the lowest order of infinities, but some infinites are demonstrably bigger than others. Also some that one would naively guess to exhibit a size difference (be of different orders) are, in fact the same size, or are members of the same order.

For example, there are an infinite number of points in the entire x, y plane such as the points (4, 9) and (1234, 5678) but there are just as many points on the x axis line, or on the x-axis, only between the points (0,0) and (1,0)! That is, the infinity of different points on the x-axis line segment between (0, 0) and (1, 0) is just as large as the infinity of points in the entire x y plane.

You have some errors inyour concepts about calculus. It is not and approximation.
The numerical values are. That's why they are called "limits". It just calculates which numer it is "heading to".

For example the intergral of x is exacly (x^2)/2, not approximately this. In this general form there is no numerical value. If instead you want o speak of the "definite intergal" of x between 1 and 2, there is a numerical value: (4 -1)/2 = 3/2, exactly or 1.5 exactly, not "approximately."
Well, of course, because definite integrals don't deal with infinity. You use them to try to find a value between two definite points.

Now some of calculus resullts (not calculus itrself) can be impossible to write in numerical form such as 1.5 is in numerical form. For example, the square root of 2 could be the result of some calculus computation. You know that any attempt to write it in numerical form, such as 1.414213....... , is only an approximation, but this is not "calculus's fault." - If a calculus calculation of this type produces the square root of 2, the it produced exaclty the correct answer.
Yes, I know that. I never said it was "calculus fault", I said it produces an aproximation.

Unfortunately, some calculus integrations are impossilbe to do (I think that may be true, but many that are certainly impossible for me, which may yield up their answers to some one else. Perhaps all might yield to some one smarter, more clever than any human. I.e. I am not enough of a mathematician to know if it has been proven that some well defined (analytic) functions can not be analytically integrated or not. Often complex integrands can not in practice be analytically integrated, so then if you still want an answer, you must result to "numerical integration" which is onlly approximate.
Yeah... I never understood why some of them can't give us an answer. Maybe the answer doesn't exist in this universe..... :D

I think this because definition (1) clearly relates to only the lowest order of infinities, but some infinites are demonstrably bigger than others.
Check your logic. (1) states that infinity is a value greater then any specific value. Then you say that one infinity may be greater then the other. If one "infinit"y is smaller, then that "infinity" is necessarily specific. You cannot have an infinity which is greater then another because by definition, infinity is a value which is greater then any possible value.

For example, there are an infinite number of points in the entire x, y plane such as the points (4, 9) and (1234, 5678) but there are just as many points on the x axis line, or on the x-axis, only between the points (0,0) and (1,0)! That is, the infinity of different points on the x-axis line segment between (0, 0) and (1, 0) is just as large as the infinity of points in the entire x y plane.
Absolutely. But let's not go on about Zeno's Paradox here...
The reason why there are infinite numbers between two definite values is because there is no natural measure of numbers. Numbers are just conventional. I can draw a line and call the first number 1 and the other 8 and I can draw the same line with the same lenght and call the numbers 1 and the other 5. That's why we have standards of measure such as centimeters and miles. There is no such thing in nature. Nature just is. YOu cannot measure nature in nature's way because nature doesn't measure itself. It is not its purpose.

This is the reason why we will likely never find the "atom". There will always be smaller and smaller units of space-time. You can call them monads or quanta, but in the end, they are never the same.

The numerical values are. That's why they are called "limits". It just calculates which numer it is "heading to".

But they 'head to' that number and no other number. If it were an approximation ("an imprecise solution or result that is adequate for a defined purpose" (http://en.wiktionary.org/wiki/approximation)) you might say "it's heading to 2 ... or maybe 2.1" but with calculus you can clearly rule out all results but one.

At least one analytic function exists, which can not be analytically integrated.

e^(x^2) dx

But they 'head to' that number and no other number. If it were an approximation ("an imprecise solution or result that is adequate for a defined purpose" (http://en.wiktionary.org/wiki/approximation)) you might say "it's heading to 2 ... or maybe 2.1" but with calculus you can clearly rule out all results but one.
That's not what I said.
What I said is that calculus give you the number it is heading to (the limit). So you get an answer which is not the approximation. but when you deal with an infinite number, then it is simply not defined. In trigonometry, it is even easier to see that...

To Truthseeker:

Rather than defend error, get a simple calculus book. It is clear you do no know any. Your posted 6.99999999999999999999999... , only reflects ignorance about calculus, (Your comment apply to calculators, not calculus, are you confusing them?).

Your recent comments about "limits" being the reason for calculus being only an approximation are doubly wrong. First you are confusing the limiting process, which is useful in some applications, but again nothing to do with calculus. The limit of calculus are not called limits for reason you state. It is more like me saying the the "limits" of my flight are Sao Paulo airport to NYC airport.

Second reason, which your erroneously drag up in defense of earlier error, is that the approximate nature of calculus would be apparent if the limit were infinity. This also is wrong. For example, gravity is an inverse square law. That is, the force is proportional to 1/(r^2) where r is the separation of the two bodies attracting each other by gravity. If they are, as in my first post trying to help you, 1 unit apart and you integrate ALL THE WAY TO INFINITY, that is from limit 1 to the limit of infinity, then the result is very well behaved, finite and EXACT. - This is the reason that it is possible to speak of the gravitational potential. This integral is a constant time the function 1/r. When you put in the upper limit, infinity, that part of the result vanishes. (All higher inverse power laws also integrated to the upper limit infinity have finite, EXACT results.)

It would have been easy - one short paragraph to show you that only a finite amount of energy can permit the universe expand for ever, despite their being mutual gravitation attraction forever, if I had used calculus, but based on several of your prior, post, and the fact you even asked the question about expanding forever, I knew you had essentially no proficiency with calculus, so I devised an non-calculus approach to answer one of your questions for you, but now that you keep defending error, making more in doing so, it is clear you do not even know what calculus is.

If any calculus problem can be integrated (analytically, not numerically), the results are exact. You are spouting more nonsense in defense of your original false claim, producing more errors. Stop. - Get an "introduction to calculus" book and read, before you expose your ignorance more.

I hasten to add that it is no shame to be ignorant of calculus. Everyone is at some point in their lives and most remain so their entire lives, but do not continue to guess what calculus is or more post errors about it.

I don't know if it makes it any simpler, but an equation reached by the methods of calculus is close to a difference of one over infinity. The bits that are thrown out are where we reduce some of the factors to one over infinity squared or one over even higher powers of infinity.

Your recent comments about "limits" being the reason for calculus being only an approximation are doubly wrong. First you are confusing the limiting process, which is useful in some applications, but again nothing to do with calculus. The limit of calculus are not called limits for reason you state. It is more like me saying the the "limits" of my flight are Sao Paulo airport to NYC airport.

Calculus is defined analytically in terms of limits - e.g. the derivative of f at x is
lim (h->0) (f(x+h)-f(x))/h
But most people don't need to think of it this one once they know the 'higher level' rules, eg derivative of x^a is a*x^(a-1)...

Perhaps he's thinking of irrational numbers as being 'inexact'?

Calculus is defined analytically in terms of limits - e.g. the derivative of f at x is lim (h->0) (f(x+h)-f(x))/h
But most people don't need to think of it this one once they know the 'higher level' rules, eg derivative of x^a is a*x^(a-1)... It is true that a limiting process is used originally to define the foundations of differentiation, but as you observe, no one uses that after they are proficient.

It is clear from my posts, that I was speaking of integration, not differentiation, but I may not have made that as clear as possible.

Although it is not normally done, one could avoid the standard origin of differentiation as a limiting process, by defining the derivative of a function as the one which produces (by integration) the given function, sort of in analogy with the square root defined as the function which produces (by squaring) the given function. I.e. Both are "TWO WAY STREETS." (or inversion of the other)

BTW, the function you gave (I think as an example of one that could not be integrated analytically) I believe could be, but I did not reply as I have not done much of this in last 25 years, so I am rusty, not sure of myself anymore. - I think it would easily yield to the “chain rule” and substitution z = x^2 so that dz = 2xdx . Thus your function is then:

(e^z)/2x = (e^z)/2(z^½)

But even if this does not permit easy integration, I was not asking for a function that no one can integrate analytically but for a proof that such a function exist. Perhaps here also I did non make my point crystal clear. List of a 1000 functions with no known analytical integration is not proof that even one is not integrable analytically - perhaps all are, only no one has yet been clever enough to do it.

BTW, the function you gave (I think as an example of one that could not be integrated analytically) I believe could be

Hmm, I just had a look at the wikipedia integration article (http://en.wikipedia.org/wiki/Integral) and realised that 'analytically integrable' depends entirely on your definition of analytic :p for example, as you show you can convert it to 2*z^(-1/2)*e^z whose integral is just a generalised gamma function. Is that analytic? Given any function f you can define g as the integral of f, which doesn't mean g isn't a function even if you can't write it in terms of ln, sin, exponents etc.

You can define differentiation as 'anti-integration' (although differentiation is usually taught first since it's easier) but even integration may use limits in its proof (http://en.wikipedia.org/wiki/Lebesgue_integral#Proof_techniques). But I don't see that as a problem - limits are a useful tool, not just an 'aproximation' to be avoided for some philosophical reason :D

...Given any function f you can define g as the integral of f, which doesn't mean g isn't a function even if you can't write it in terms of ln, sin, exponents etc....again i agree with you. I do not know the definition of analytic, but presume there is an adequate one. If I had to make one up, designed to reject your argument above, I would say that the "analytic result" of an integration is experssible in terms of functions and constants (like e & pi) which have been found in several other cases, not related to the particular funcion or integrand being cosidered. I.e. functional forms of sufficient general use and interest that at least some obsure mathematician has worked out either a table of their values or some polinomial expansion for calculating them etc.

But we are getting far from the thread, and unfortunately the relative few people active insciforums who actually know much more than me probably do not read this thread. If I can find one where they are active, and open to slight diversion, I will try to ask if there is a proof that some "analytic" functions do not have analytic integrations. (and learn what "analytic" really means too, I hope.)

To Truthseeker:

Rather than defend error, get a simple calculus book.
Blah blah blah blah blah blah...
That's the best you can do when you make a mistake.

Your posted 6.99999999999999999999999... , only reflects ignorance about calculus, (Your comment apply to calculators, not calculus, are you confusing them?).
Calulators and calculus... haha... :p

First you are confusing the limiting process, which is useful in some applications, but again nothing to do with calculus.
Of course it has nothing to do with calculus! It's just the BASIS of it. :bugeye:

The rest is blah blah blah blah...
I suppose my grade A in calculus is completely irrelevant, eh....? :rolleyes:

But we are getting far from the thread, and unfortunately the relative few people active insciforums who actually know much more than me probably do not read this thread. If I can find one where they are active, and open to slight diversion, I will try to ask if there is a proof that some "analytic" functions do not have analytic integrations. (and learn what "analytic" really means too, I hope.)
That's not the purpose of the thread...