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Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is certainly Taylor expansion. They have a quantity (function) that they need to approximate: they expand it in Taylor series, keep the order of approximation that is useful for their purposes, and discard the irrelevant terms.

Appearently, there is little preoccupation for mathematically justifying this procedure, even if the to-be-approximated quantity is not given by an explicit form which is clearly known to be analytic. As Physics clearly gets no problems from the above mathematical subtleties, this may just mean that the distinction between analytic and smooth functions is somehow irrelevant to the basic equations of physics, or rather to the approximations of their solutions that are empirically testable.

If non-analytic smooth functions are irrelevant to Physics, why is it so?

 

Are there equations of physical importance in which non-analytic smooth solutions actually are important and cannot be safely considered "as if they were analytic" for the approximation purposes?

Remark: analogous questions may arise about Fourier series expansions.

One possible way the practice goes might be:

1. Consider a (differential or otherwise) equation $P(f)=0$ usually with analytic coefficients.
2. Expand the coefficients in Taylor series around a point in the scale of physical interest.
3. Discard higher order terms obtaining an approximated equation with polynomial coefficients $\tilde{P}(f)=0$.
4. Make the ansatz that the solutions $f$ of interest must be analytic.
5. Find the coefficients of $f$ by hand or by other means.

This leaves open the question why the ansatz is mathematically justified, if the equation of interest was $P$ not $\tilde{P}$. Do analytic solutions of $\tilde{P}$ aptly approximate solutions of $P$? Edit: I understand now that these last two lines are not very well formulated. Perhaps, ignoring the $\tilde{P}$ thing, I should have just asked something like:

Given any $\epsilon>0$, does knowing the analytic solutions (i.e. knowing their coefficients, possibly up to an arbitrarily large but finite number of digits) of $P$ give all the information about all solutions of $P$ up to $\epsilon$-approximation? Are there physically well known classes of equations $P$ in which this may not happen (perhaps even up to taking very regular approximations of the coefficients/parameters of $P$ itself)?

Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is certainly Taylor expansion. They have a quantity (function) that they need to approximate: they expand it in Taylor series, keep the order of approximation that is useful for their purposes, and discard the irrelevant terms.

Appearently, there is little preoccupation for mathematically justifying this procedure, even if the to-be-approximated quantity is not given by an explicit form which is clearly known to be analytic. As Physics clearly gets no problems from the above mathematical subtleties, this may just mean that the distinction between analytic and smooth functions is somehow irrelevant to the basic equations of physics, or rather to the approximations of their solutions that are empirically testable.

If non-analytic smooth functions are irrelevant to Physics, why is it so?

Are there equations of physical importance in which non-analytic smooth solutions actually are important and cannot be safely considered "as if they were analytic" for the approximation purposes?

Remark: analogous questions may arise about Fourier series expansions.

One possible way the practice goes might be:

1. Consider a (differential or otherwise) equation $P(f)=0$ usually with analytic coefficients.
2. Expand the coefficients in Taylor series around a point in the scale of physical interest.
3. Discard higher order terms obtaining an approximated equation with polynomial coefficients $\tilde{P}(f)=0$.
4. Make the ansatz that the solutions $f$ of interest must be analytic.
5. Find the coefficients of $f$ by hand or by other means.

This leaves open the question why the ansatz is mathematically justified, if the equation of interest was $P$ not $\tilde{P}$. Do analytic solutions of $\tilde{P}$ aptly approximate solutions of $P$? Edit: I understand now that these last two lines are not very well formulated. Perhaps, ignoring the $\tilde{P}$ thing, I should have just asked something like:

Given any $\epsilon>0$, does knowing the analytic solutions (i.e. knowing their coefficients, possibly up to an arbitrarily large but finite number of digits) of $P$ give all the information about all solutions of $P$ up to $\epsilon$-approximation? Are there physically well known classes of equations $P$ in which this may not happen (perhaps even up to taking very regular approximations of the coefficients/parameters of $P$ itself)?

Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is certainly Taylor expansion. They have a quantity (function) that they need to approximate: they expand it in Taylor series, keep the order of approximation that is useful for their purposes, and discard the irrelevant terms.

Appearently, there is little preoccupation for mathematically justifying this procedure, even if the to-be-approximated quantity is not given by an explicit form which is clearly known to be analytic. As Physics clearly gets no problems from the above mathematical subtleties, this may just mean that the distinction between analytic and smooth functions is somehow irrelevant to the basic equations of physics, or rather to the approximations of their solutions that are empirically testable.

If non-analytic smooth functions are irrelevant to Physics, why is it so?

Are there equations of physical importance in which non-analytic smooth solutions actually are important and cannot be safely considered "as if they were analytic" for the approximation purposes?

Remark: analogous questions may arise about Fourier series expansions.

One possible way the practice goes might be:

1. Consider a (differential or otherwise) equation $P(f)=0$ usually with analytic coefficients.
2. Expand the coefficients in Taylor series around a point in the scale of physical interest.
3. Discard higher order terms obtaining an approximated equation with polynomial coefficients $\tilde{P}(f)=0$.
4. Make the ansatz that the solutions $f$ of interest must be analytic.
5. Find the coefficients of $f$ by hand or by other means.

This leaves open the question why the ansatz is mathematically justified, if the equation of interest was $P$ not $\tilde{P}$. Do analytic solutions of $\tilde{P}$ aptly approximate solutions of $P$? Edit: I understand now that these last two lines are not very well formulated. Perhaps, ignoring the $\tilde{P}$ thing, I should have just asked something like:

Given any $\epsilon>0$, does knowing the analytic solutions (i.e. knowing their coefficients, possibly up to an arbitrarily large but finite number of digits) of $P$ give all the information about all solutions of $P$ up to $\epsilon$-approximation? Are there physically well known equations/theories in which this doesn't happen?

Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is certainly Taylor expansion. They have a quantity (function) that they need to approximate: they expand it in Taylor series, keep the order of approximation that is useful for their purposes, and discard the irrelevant terms.

Appearently, there is little preoccupation for mathematically justifying this procedure, even if the to-be-approximated quantity is not given by an explicit form which is clearly known to be analytic. As Physics clearly gets no problems from the above mathematical subtleties, this may just mean that the distinction between analytic and smooth functions is somehow irrelevant to the basic equations of physics, or rather to the approximations of their solutions that are empirically testable.

If non-analytic smooth functions are irrelevant to Physics, why is it so?

Are there equations of physical importance in which non-analytic smooth solutions actually are important and cannot be safely considered "as if they were analytic" for the approximation purposes?

Remark: analogous questions may arise about Fourier series expansions.

One possible way the practice goes might be:

1. Consider a (differential or otherwise) equation $P(f)=0$ usually with analytic coefficients.
2. Expand the coefficients in Taylor series around a point in the scale of physical interest.
3. Discard higher order terms obtaining an approximated equation with polynomial coefficients $\tilde{P}(f)=0$.
4. Make the ansatz that the solutions $f$ of interest must be analytic.
5. Find the coefficients of $f$ by hand or by other means.

This leaves open the question why the ansatz is mathematically justified, if the equation of interest was $P$ not $\tilde{P}$. Do analytic solutions of $\tilde{P}$ aptly approximate solutions of $P$? Edit: I understand now that these last two lines are not very well formulated. Perhaps, ignoring the $\tilde{P}$ thing, I should have just asked something like:

Given any $\epsilon>0$, does knowing the analytic solutions (i.e. knowing their coefficients, possibly up to an arbitrarily large but finite number of digits) of $P$ give all the information about all solutions of $P$ up to $\epsilon$-approximation? Are there physically well known classes of equations $P$ in which this may not happen (perhaps even up to taking very regular approximations of the coefficients/parameters of $P$ itself)?