L{erf(t)} = 1 / (s * sqrt(s + 1))

L{erf(t)}=e^s^2(1−erf(s))​/s

a proof

https://proofwiki.org/wiki/Laplace_Transform_of_Error_Function

You may look at No. 4 and 7 for both erf and erfc

https://eqworld.ipmnet.ru/en/auxiliary/inttrans/laplace7.pdf

yes, your derivation is correct

Hi ..can someone kindly share the derivation..that will be more useful.

The problem was likely to be a sign error or that you skipped over the subtraction structure of the complimentary error function. The first derivation you came with involves a more complicated integral representation than the complementary error function . The correct L{erf(a/√t)} is (1 - e^(-2 sqrt(s) abs(a)))/s, your teacher's result.

By the way, 1/k(1-exp(-√ak))) is the transformation mentioned in EqWorld

Mohit Rattanpal, you can see a complete derivation for complementary error function, erfc, at

https://www.freemathhelp.com/forum/threads/laplace-transform-of-error-function-laplace-transform-of-erfc-1-sqrt-t.137110/

and simply apply it in your case for error function, erf.

Or, you can follow the two steps there in:

(1) change the order of integration of a double integral, and

(2) apply the Cauchy-Schlömilch transformation for easily solving an integral Article The Cauchy-Schlomilch transformation

Good luck

Dear All..Thanks for participating in this discussion.I have attached the derivation for the L{erf (a/sqrt t)} , I would be grateful if you could review the same and point out error in it.Specifically, critically examine the integral limits transformation.

Try to fix integrals limits - when the order of integration is changed ...

and you are ready to go ...

I attempted to derive step by step. The main challenge here is to understand the reversal of the limits during change of the order of integration.

OK, you are done!

(only one negligible typo)

Good job with the differentiation under the integral sign.

Though, it's good to know the fact that

The Cauchy-Schlomilch transformation states that for a functionf and a, b > 0, the integral of f(x^2) and a*f((ax − b/x)^2) over the interval[0, ∞) are the same

... and, the other similar fact.

Steftcho P. Dokov thanks for pointing out.