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edited Aug 10, 2016 at 18:35

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Automatically SubbingSubstituting In A Series Into An Integral

I'm working on an algorithm that includes some difficult integration, dealing with $erf(t)$$\mathrm{erf}(t)$ functions, for example. One term in the algorithm is this:

Integrate[Exp[-2 t^2] t^2 Erf[t], {t, 0, x}]

which integrates just fine. Next, we have another term:

Integrate[Exp[-2 t^2] t^2 Erf[Sqrt[3] t], {t, 0, x}],

where we have added the $\sqrt{3}$ in the argument of the $erf(t)$$\mathrm{erf}(t)$. MathematicaMathematica will integrate this if we approximate either $exp(-2 t^2)$$\exp(-2 t^2)$ or $erf(\sqrt{3}t)$$\mathrm{erf}(\sqrt{3}t)$ as Maclaurin series, for example.

What I want MathematicaMathematica to do is if it can't do an integral that includes an $erf(t)$$\mathrm{erf}(t)$ function, then it would sub in Maclaurin series for the function and try again. I don't want to sub in the series prematurely (for all instances of $erf(t)$$\mathrm{erf}(t)$) because the series are not that great and I'm using them more as a crutch. Since these integrations are automated (they are part of a DoDo loop), I can't do this substitution manually.

If there is an option that already exists, or that could be created that would accomplish this, I would be over the moon.

Thank you!

Automatically Subbing In A Series Into An Integral

I'm working on an algorithm that includes some difficult integration, dealing with $erf(t)$ functions, for example. One term in the algorithm is this:

Integrate[Exp[-2 t^2] t^2 Erf[t], {t, 0, x}]

which integrates just fine. Next, we have another term:

Integrate[Exp[-2 t^2] t^2 Erf[Sqrt[3] t], {t, 0, x}],

where we have added the $\sqrt{3}$ in the argument of the $erf(t)$. Mathematica will integrate this if we approximate either $exp(-2 t^2)$ or $erf(\sqrt{3}t)$ as Maclaurin series, for example.

What I want Mathematica to do is if it can't do an integral that includes an $erf(t)$ function, then it would sub in Maclaurin series for the function and try again. I don't want to sub in the series prematurely (for all instances of $erf(t)$) because the series are not that great and I'm using them more as a crutch. Since these integrations are automated (they are part of a Do loop), I can't do this substitution manually.

If there is an option that already exists, or that could be created that would accomplish this, I would be over the moon.

Thank you!

Automatically Substituting In A Series Into An Integral

I'm working on an algorithm that includes some difficult integration, dealing with $\mathrm{erf}(t)$ functions, for example. One term in the algorithm is this:

Integrate[Exp[-2 t^2] t^2 Erf[t], {t, 0, x}]

which integrates just fine. Next, we have another term:

Integrate[Exp[-2 t^2] t^2 Erf[Sqrt[3] t], {t, 0, x}],

where we have added the $\sqrt{3}$ in the argument of the $\mathrm{erf}(t)$. Mathematica will integrate this if we approximate either $\exp(-2 t^2)$ or $\mathrm{erf}(\sqrt{3}t)$ as Maclaurin series, for example.

What I want Mathematica to do is if it can't do an integral that includes an $\mathrm{erf}(t)$ function, then it would sub in Maclaurin series for the function and try again. I don't want to sub in the series prematurely (for all instances of $\mathrm{erf}(t)$) because the series are not that great and I'm using them more as a crutch. Since these integrations are automated (they are part of a Do loop), I can't do this substitution manually.

If there is an option that already exists, or that could be created that would accomplish this, I would be over the moon.

Thank you!

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asked Aug 10, 2016 at 18:30

Buddhapus

Automatically Subbing In A Series Into An Integral

I'm working on an algorithm that includes some difficult integration, dealing with $erf(t)$ functions, for example. One term in the algorithm is this:

Integrate[Exp[-2 t^2] t^2 Erf[t], {t, 0, x}]

which integrates just fine. Next, we have another term:

Integrate[Exp[-2 t^2] t^2 Erf[Sqrt[3] t], {t, 0, x}],

where we have added the $\sqrt{3}$ in the argument of the $erf(t)$. Mathematica will integrate this if we approximate either $exp(-2 t^2)$ or $erf(\sqrt{3}t)$ as Maclaurin series, for example.

If there is an option that already exists, or that could be created that would accomplish this, I would be over the moon.

Thank you!

calculus-and-analysis