Complex Integral to error function

[WWCY]

Homework Statement

I have an integral

$$\int_{-\infty}^{0} e^{-(jp - c)^2} \ dp$$

where j and c are complex, which I'd like to write in terms of $\text{erf}$

I'd like to know what would happen to the integral limits as I make the change of variables $t = jp - c$.

1) As $p$ tends to negative infinity, am I allowed to write the lower limit of the integral simply as $-\infty$?

2) When $p = 0$, the upper limit becomes $t = -c$, which is a complex number. Does this mean that I am unable to write the integral as an error function?

Many thanks in advance.

Homework Equations

The Attempt at a Solution

 

[Ray Vickson]

Homework Statement

I have an integral

$$\int_{-\infty}^{0} e^{-(jp - c)^2} \ dp$$

where j and c are complex, which I'd like to write in terms of $\text{erf}$

I'd like to know what would happen to the integral limits as I make the change of variables $t = jp - c$.

1) As $p$ tends to negative infinity, am I allowed to write the lower limit of the integral simply as $-\infty$?

2) When $p = 0$, the upper limit becomes $t = -c$, which is a complex number. Does this mean that I am unable to write the integral as an error function?

Many thanks in advance.

Homework Equations

The Attempt at a Solution

In you change the variable to $t = jp-c$ the limits are $-j \, \infty-c$ and $j\, 0 -c$, so you are integrating along a straight line parallel to the imaginary $t$-axis.

You can certainly write the integral in terms of the error function (depending on exactly how you define $\text{erf}(\cdot)$).

 

[WWCY]

Thank you for your response!

In you change the variable to $t = jp-c$ the limits are $-j \, \infty-c$ and $j\, 0 -c$, so you are integrating along a straight line parallel to the imaginary $t$-axis.

You can certainly write the integral in terms of the error function (depending on exactly how you define $\text{erf}(\cdot)$).

The only $\text{erf}(x)$ I know of is $\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} \ dt$, but this seems to only take real limits (unless I am mistaken). Is it possible, then, to write it in this form? If not, which form of error function should I use?

Thanks very much for your assistance.

 

[Ray Vickson]

Thank you for your response!
The only $\text{erf}(x)$ I know of is $\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} \ dt$, but this seems to only take real limits (unless I am mistaken). Is it possible, then, to write it in this form? If not, which form of error function should I use?

Thanks very much for your assistance.

The function $\text{erf}(z)$ for complex $z$ is defined just by replacing $x$ by $z$ in the integral above. See, eg.,
http://mathworld.wolfram.com/Erf.html

Of course, one needs to pick a path from $0$ to $z$, but the integral is independent of the chosen path, because $\exp(-t^2)$ is analytic in the complex $t$-plane.

 

[Ray Vickson]

Thank you for your response!
The only $\text{erf}(x)$ I know of is $\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} \ dt$, but this seems to only take real limits (unless I am mistaken). Is it possible, then, to write it in this form? If not, which form of error function should I use?

Thanks very much for your assistance.

You should write it as $\lim_{N \to \infty} I_N,$ where
$$I_N = \int_{-N}^0 e^{-(j p - c)^2} \, dp,$$
and then worry about whether $I_N$ has a finite limit as $N \to +\infty.$

 

[WWCY]

You should write it as $\lim_{N \to \infty} I_N,$ where
$$I_N = \int_{-N}^0 e^{-(j p - c)^2} \, dp,$$
and then worry about whether $I_N$ has a finite limit as $N \to +\infty.$

I'm not sure I follow what you're trying to say here. If I leave the integral as it is, how do I evaluate it analytically without use of the erf?

The function $\text{erf}(z)$ for complex $z$ is defined just by replacing $x$ by $z$ in the integral above. See, eg.,
http://mathworld.wolfram.com/Erf.html

Does this mean that erf is able to take and produce complex values (i.e. $\text{erf} (x + iy)$)?

If so, is there a way to write a complex erf in real and imaginary parts? I ask as I'm interested in calculating the absolute value of the complex erf (or the original integral to be exact).

Thank you for your patience and assistance.

 

[Ray Vickson]

I'm not sure I follow what you're trying to say here. If I leave the integral as it is, how do I evaluate it analytically without use of the erf?
Does this mean that erf is able to take and produce complex values (i.e. $\text{erf} (x + iy)$)?

If so, is there a way to write a complex erf in real and imaginary parts? I ask as I'm interested in calculating the absolute value of the complex erf (or the original integral to be exact).

Thank you for your patience and assistance.

Yes. you can compute $\text{erf}(x+iy)$. One way would be to use the series given in the link I provided. Most computer algegra systems (such as Mathematica or Maple---the one I use) can do it easily. For example, here is a screen shot of what I get using Maple (which uses I for $\sqrt{-1}$):

r:=erf(2+3*I); <---input
r := erf(2 + 3 I) <-- echoed output

> evalf(r); <-- input --- means "floating-point evaluation

-20.82946143 + 8.687318271 I <---- output at standard default accuracy

> evalf[50](r);

-20.829461427614568389103088451981112874439035666354 +

8.6873182714701631444280787545418715530519896486487 I <----- 50 digit accuracy

Anyway, the definition of an improper integral such as yours is
$$\int_{-\infty}^0 f(p) \, dp = \lim_{N \to \infty} \int_{-N}^0 f(p) \, dp.$$
So, I ask again: how do you know if a limit exists? Certainly the integral is expressible in terms of "erf" for finite $N$, but it will involve $\text{erf}(c-iN)$, so an argument having a large imaginary part. Does that have a finite limit? Can you say for sure that the answer does not have the form $\pm \infty \pm i \infty?$

 

[WWCY]

So, I ask again: how do you know if a limit exists? Certainly the integral is expressible in terms of "erf" for finite $N$, but it will involve $\text{erf}(c-iN)$, so an argument having a large imaginary part. Does that have a finite limit? Can you say for sure that the answer does not have the form $\pm \infty \pm i \infty?$

I'm not sure I'm able to answer that as the integrand I'm considering is quite complicated. However, this is a smaller problem in the context of a larger issue and I might have taken an unnecessarily complicated route. I'll start another thread writing the problem from scratch.

Thank you so much for your assistance and apologies for the inconvenience.

 

[WWCY]

*edit: j in the original post refers to some complex number. Apologies for the poor notation.

 

[Ray Vickson]

*edit: j in the original post refers to some complex number. Apologies for the poor notation.

So, we have
$$A=\int_{-\infty}^0 e^{-[(a+ib)p - (r+is)]^2} \, dp, $$
where $a,b,r,s$ are real constants.

If $b = 0$ and $a \neq 0$ you should have no problem expressing the answer in terms of "erf", and the answer $A$ is decidedly finite. However, if $b \neq 0$ you could potentially be in trouble, and the convergence issue raises its ugly head. I won't tell you the answer, just tell you to be careful.