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  1. The x+y function that appears to just be a plane? Do you think that's not invertible in any way? When a=0, the integral gives you a constant line which when inverted gives you simply a vertical line, and to make it functional it only needs to be equivalent to a translated Dirac delta function or limit the range to a finite number. Literally anything that you can recognize as any kind of curve or shape can be flipped over the line y=x, that's really all you need to ask yourself, the rest is simply a matter of domain/range restrictions and whether or not it's in terms of known functions. Matrices are their own group, it has it's own rules. The definition for a bisection or an inversion or invertible does not necessarily have all the same applications and versatility as in functional analysis. Is there any particular reason you continue to completely evade the specific example I provided wherein the end result is already known but it's complex enough to illustrate the application? Honestly this site is so behind, stackexchange and physicsforums answer things so much quicker and with more detail compared to this site and without the cavalier attitude of the academically disappointing precedent set by the staff. As an allegedly accredited staff member you should be able to view the edit history, and if you did you'd see I added the example only shortly after the post in order to give context and a more identifiable goal. Do you realize this looks like abuse and thus a lack of integrity? Uh, no one's forcing you to be here, as a sentient being, your frustration is your own choice. If you don't have the expertise or experience for making progress with this particular technique, go ahead and do something else.
  2. Your failure to find in inverse comes purely from your choice to refrain from defining a branch, something that allows any function to be invertible and isn't limited to functional analysis but is applicable to linear algebra as well. Not only that, but this is not the example I provided. Anything that you can possibly identify in any way as any kind of continuous curve or line is invertible over that specific segment, it is only a matter of whether or not you apply the proper techniques to find it in terms of identified functions. In your own limited capabilities, y=x^2 shouldn't actually have an inverse, and yet every actual accredited mathematician in the world agrees that http://www.wolframalpha.com/input/?i=inverse+y%3Dx%5E2 And when it doesn't know the answer, it doesn't say "never going to be an answer, no one will ever know just give up on math" it says "no results in terms of *standard* mathematical functions" because accredited mathematicians know that a function is invertible if you apply the technique of choosing the proper domain and range, it is only the case that we may lack the techniques to explicitly identify it.
  3. No the answer to momentum will depend on a vector analysis of the situation and whether the system is opened or closed. Like I said, every function can be inverted to some degree if the right domain is chosen, whether or not it happens over a particular domain isn't your problem, the topic you derailed is in reference to an example where we already know what the outcome should be.
  4. I garuntee you there is a technique that you can attempt to apply to the situation I provided that can also work in similar situations. If I choose to apply the technique to some random problem and I find the result is not expressible in currently defined functions, that's my problem, not yours. From what I have seen, it is exceptionally rare to get an integral definition that somehow turns out to be nice and easy to work with in some alternate definition. In terms of functions you know of, or, have cared to define. What the question and example asks for the technique to show how it's done to verify a result, or in other words, any amount of evidence to verify any of the "answers" thus far.
  5. It is true unless it is completely common knowledge that it's already been proven. Nope, when the answer is well-known enough, that's the rule, not broad enough. If you ask a broad question like "what's the inverse of a second degree polynomial?" "sometimes where x=0" isn't going to suffice, you need the quadratic formula at the very, most basic east. And you don't need to, it's very clear that it is. I also specifically asked about a "function that is defined as an integral" and specifically gave a specific example to work through, "In fact, you know what, let's take an example: arctangent. The arctangent function can be described as the integral of 1/sqrt(1+x^2) which after complex analysis or trig substitution we know actually takes the form of a complicated complex-valued logarithm containing a polynomial. But now, let's say I wanted to define the actual regular tangent function just starting out with the integral definition of the arctangent function (which for verification purposes we know takes the form of a sum/quotient of real/imaginary exponents)..." If it's a function that is continuous over some interval and you isolate a branch point, which is the only time you could ask for an inverse, the answer is never ever "sometimes" or "no." Good thing it's not impossible to tell then. Even the gamma function with 100% certainty has an inverse over any branch point you choose, at the very least over the interval that it is monotonically increasing, it is simply the basic, common, average case that the inverse it is not currently defined in terms of elementary functions or possibly just known techniques, outside of a taylor series. This comment of yours is off topic.
  6. Incorrect. In math an science, the answer isn't some arbitrary whim, an answer needs to include evidence and reasoning or it isn't credible and therefore is unusable. Off topic from mathematics, and I will continue to report that content until it is censored as per the rules of the site.
  7. Not only is that a useless tautology which once again illustrates my point, but most known integral functions absolutely have an inverse over a large domain even if it's only known by it's taylor series, so now what you're saying is also completely irrelevant, it's as if you're just here to troll. If you're not going to take the time to answer the question accurately, don't waste the world's time with your post.
  8. This does not help anyone. The correct response would be to start looking at the inverse function theorem which definitely has something to do with the problem at hand.
  9. defining composite functions and inverses

    Why? Because the foundation of math and science isn't in automatically assuming every single thing you think of is automatically correct. I'm not just going to make a statement and assume it's going to work out, I'm going to test a lot of times, I only asked it here in case I didn't have luck finding out the right answer or to see if anyone else could confirm the results.
  10. If I have a given function that is defined as an integral, like of something that's clearly hard to work with like the gamma function or the exponential integral or the cosine integral or the error function or etc, is there a way to define an inverse of that function in a similar form that's NOT a cop-out dy integral? Not some dy integral that forces me to switch to integrating along the y-axis for no reason that solves absolutely nothing and never will, but something that starts as f(x) = integral(g(x))dx and hopefully takes the form of f-1(x) = integral g^-1(x)dx or something similar, something that is still defined on the domain of x. In fact, you know what, let's take an example: arctangent. The arctangent function can be described as the integral of 1/sqrt(1+x^2) which after complex analysis or trig substitution we know actually takes the form of a complicated complex-valued logarithm containing a polynomial. But now, let's say I wanted to define the actual regular tangent function just starting out with the integral definition of the arctangent function (which for verification purposes we know takes the form of a sum/quotient of real/imaginary exponents)...
  11. Determining the relationship to elementary functions?

    I know there's an answer to this, because if there wasn't, every single person who ever said "cannot be put into terms of elementary functions" is automatically discretited, so I know they definitely must have done something to verify that.
  12. defining composite functions and inverses

    Yeah I've found you're not entirely correct. You're obviously correct that not every function is invertable on every domain, but as a general technique it absolutely works because it's the same as doing a u-substitution and then back substituting it to find the inverse. I've done it tons of times now and it works every single time, long as I limit the domain.
  13. Well you're in luck then because there's nothing to believe. Where does the statement 1+1=2 say anything about time or space? It's not a question of whether or not it physically exists, it's just a matter of whether or not a logical correlation is true, and every one is true regardless of any position of anything or any amount of time that passes. No matter where or when you are in the universe, 1+1=2 is a true statement within any set of axioms that allow that statement to be true. For something closer to home, take for instance a dimension: we can't pick it up, we can't see one, but it is nonetheless the consensus in physics that dimensions exist. As you said, a lack of defined natural numbers would imply a lack of uniqueness. What I'm asking is that if you defined operators like succession but didn't define uniqueness, it seems like there would be nothing to verify that 2=3 is false, even in your point that you could disprove the statement with succession since that relies on mathematics that has already been established. So the symbols 2 and 3 could be equal if there's nothing explicitly to say that any given number x and y (or a and b) have to be unique values. This isn't philosophy this is more on the main topic.
  14. Logically speaking, logic itself is not dependent on any amount of time nor any amount or space. Even if the universe didn't exist, 1+1=2 would still be a true statement relative to some immaterial set of axioms. But Shakespeare's play requires a physical universe to exist based on its very definition. That's true to some extent, but what I am referring to is ruling out the possibility that one value is equal to another value, just because you write different symbols in this "symbolic game." It seems like without uniqueness, 2=3 could actually be a true statement. And as you can imagine, even if an element of a set is succeeded by another element of a set, it doesn't seem like you could garuntee the values are unique when dealing with complex or imaginary numbers, or even just angles. You can't even say that one complex imaginary number is greater than another.
  15. Well symbols aside, logic is logic regardless of whether we discover it or not. All logical correlations are true all the time. What if a particular set of axioms lacked a specific statement of uniqueness? Could you still prove it?