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  1. We all know that roots of polinomials are not always real numbers or when we take the square root of a negative number, we need to immmediately define an imaginary number called $i$ or $j$ for futher calculations.

  2. There is also an analogy between a polynomial and ordinary differential equations, one of which has a solutions with a number and the other with a function.

  3. According to 1. and 2. it is a strightforward question to ask if there is a definition of a function called imaginary function for futher possible computations? Or does it make sense?

EDIT: As it might be confusing with a complex function $f(a+bj)$; I want a basic function like the imaginary number $i$ as a solution to unsolvable equations and via use of this function, I want to have some solutions to other unsolvable differential equations. Like for the number $i$, and a polynomial.

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The question is a little confusing. You want a new function that is the solution to otherwise unsolvable differential equations, not an imaginary function (a function which takes imaginary values) right? – Nameless Nov 18 '12 at 12:57
@J.M. The OP is saying: Some polynomial equations don't have solutions in $\mathbb{R}$ so we constructed the field of complex numbers. Can the same be done in the field of differential equations? – Nameless Nov 18 '12 at 13:12
@SeyhmusGüngören Change the title to: "Axiomatically define a new function that can solve otherwise impossible differential equations, like $i$ solves otherwise impossible polynomial equations" or something like that. – Nameless Nov 18 '12 at 13:14
I think this might be in the intended direction: Some (e.g. differential) equation might be solvable by a formal power series with zero radius of convergence. The formal power series is thus not really a function, but can be investigated as a power series; it may also happen that further manipulations cancel away the divergence and one obtains a useful (i.e. convergent) end result. This would be a bit like the situation early in the history of imaginaries when one was happy to have a real final result even if it might involve imaginary intermediate resuls. Not sure if this makes sense. – Hagen von Eitzen Nov 18 '12 at 13:17
@SeyhmusGüngören: no, not all differential equations can be solved, for example the equation $f(x)=f(x)+1$. However, such unsolvable equations are not physically interesting. I do not know if there is some abstract branch which investigates these equations nonetheless. – akkkk Nov 18 '12 at 13:19
up vote 6 down vote accepted

The introduction of $i$ is not, as the question seems to suggest, just a matter of introducing a new symbol axiomatically. The important content of this extension of the number concept is that, after introducing $i$, one can still use most (though not all) of the calculation rules that are familiar for real numbers. More precisely, the complex numbers form a field (though not an ordered field), so we can work with addition and multiplication (but not $<$) in $\mathbb C$ just as we do in $\mathbb R$. In this light, the analogous question for differential equations should not be "can we axiomatically introduce new quasi-functions to serve as solutions" but rather "can we axiomatically introduce new quasi-functions to serve as solutions and retain important properties that we are accustomed to using for working with ordinary functions?" The crucial point, which requires real mathematical work, is to figure out what to retain (like $+$ and $\times$ in the case of $\mathbb C$) and what to give up (like $<$). Then, if we're lucky and have made good choices, we might be able to develop a theory that provides solutions for differential equations that don't admit ordinary solutions.

In fact, such theories have been developed, involving distributions, Sobolev spaces, and the like. You can probably find tons of information about these by searching for "weak solutions". One of the things one gives up is the notion (usually regarded as central to the very notion of function) of evaluating a function at a point. If $f$ is a distribution, there is often no way to assign a meaning to $f(a)$ for a specific value of $a$. Nevertheless, other things, like differentiation, make good sense on distributions.

For some types of differential equations (elliptic ones, I believe), when one gets a weak solution it "magically" turns out to be an ordinary, smooth function, despite whatever quasi-function methods were used to obtain it. This can be viewed as analogous to the case where a calculation with complex numbers "magically" leads to a real number as the final result.

For other types of equations, weak solutions are really necessary and one can't expect ordinary solutions. For example, some equations governing fluid dynamics allow for the development of discontinuities where the function or its derivatives become undefined, even if the initial conditions were perfectly smooth. This is not a defect of the equations or the solutions; rather it models a real physical phenomenon, the development of shocks.

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There is a purely algebraic theory of solutions of differential equations which may be viewed as providing the sort of differential analog that you seek. A differential field is a field with a derivation, i.e. linear map $\rm\:y\to y'$ satisfying the product rule $\rm\:(uv)' = u'v + v'u.\:$ Let $\rm\:F\:$ be a differential field of characteristic $0$, e.g. $\rm\,\Bbb C(x),\:$ and let $\rm\:L(y) = y^{(n)} + a_{n-1} y^{(n-1)}+\,\cdots\, + a_1 y' + a_0 y = 0,\:$ $\rm\, a_i\in F,\:$ be an $\rm\,n$-th order linear differential equation with coefficients in $\rm\,F.\,$ One can prove that there exists a minimal differential "splitting field" $\rm\,K\,$ containing all of the solutions of $\rm\:L(y)= 0.\:$ Indeed, $\rm\,K\,$ may be constructed from $\rm\,F\,$ simply by formally adjoining $\rm\,n\,$ $\rm\,F$-linear independent solutions $\rm\,y_1,\ldots,y_n$ of $\rm\,L(y)= 0,\,$ along with all their derivatives, yielding the differential extension field

$$\rm K\, =\, F(y_1,\ldots,y_n, y_1',\ldots, y_n',\ldots, y_1^{(n-1)},\ldots,y_n^{(n-1)})$$

Note that this field is closed under differentiation since the equation $\rm\:L(y) = 0\:$ allows us to rewrite all $\rm\:y_i^{(k)},\,\ k\ge n,\:$ in terms of lower order derivatives. This field is unique up to a differential $\rm\,F$-isomorphism and is known as the Picard-Vessoit extension of $\rm\,F\,$ associated to $\rm\:L.\:$ (Note: for simplicity, I omit some technicalities, e.g. preservation of constant subfields).

Further, there is a beautiful differential analog of Galois theory which, e.g. allows one to characterize algebraically the equations solvable in a ("louivillian") differential field obtained by successive adjunctions of exponentials, integrals or algebraic elements.

This purely algebraic theory is the foundation of constructive algorithms employed in computer algebra systems for symbolic manipulation of many common elementary functions.

For an introduction to these ideas you might find it enlightening to skim the introductory sections of Michael Singer's papers, and peruse his surveys, e.g. Formal solutions of differential equations, and An outline of differential Galois theory, and Introduction to Galois theory of linear differential equations.

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Your question, I think, is as follows ...

By generalizing the concept of "function" somehow (enlarging the collection of things that we consider to be "functions"), can we then find solutions for a broader set of differential equations. You use complex numbers as an analogy -- by generalizing real numbers, they allow a broader set of polynomial equations to be solved.

I would say that this extension/enlargement has already been done, to some extent. We have theorems that tell us that solutions exist, even though we can't compute them, or we have functions that can only be expressed as the limits of iterative processes, and so on. In some sense, these kinds of functions are "unnatural" or intangible, just as imaginary numbers are.

On the other hand, if something is going to be a solution of a differential equation, it can't be too exotic and un-function-like, it seems to me. It has to be differentiable, for one thing.

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I think every mathematician would agree we start looking for solutions in the space of continuous functions anyway - heck, we define many functions in terms of differential equations! Only the last paragraph actually relates to the question, and it is not very scientific... – akkkk Nov 18 '12 at 13:28

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