Seeking Detailed Explanation for Transforming an Integral Equation Using Euler's Formula and Error Function

Question

I am working on understanding the transformation of a specific integral equation into a simpler form using Euler's Formula and the Error Function. The original equation is: $$ u(x, t) = u_0\left\{1 - \frac{1}{\pi} \int_0^{\infty} \frac{e^{-vt}}{v} \sin \left(\left(\frac{v}{k}\right)^{\frac{1}{2}} x\right) dv\right\} $$ And it is transformed into:

$$ u(x, t) = u_0\left\{1 - \operatorname{erf}\left(\frac{x}{2 \sqrt{kt}}\right)\right\} $$

The error function is defined as:

$$ \operatorname{erf}(w) = \frac{2}{\sqrt{\pi}} \int_0^w e^{-v^2} dv $$

Additionally, I know that the sine function can be expressed using Euler's formula:

$$ \sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i} $$

Using Wolfram Mathematica, I have the following steps:

1. Initial expression: initialExpr[u0_, x_, t_, k_] := u0*(1 - (1/Pi)*Integrate[Exp[-v*t]/v*Sin[Sqrt[v/k]*x], {v, 0, Infinity}, Assumptions -> {t > 0, k > 0}])

2. Euler's formula substitution: eulerSubstitute = Sin[Sqrt[v/k]*x] -> (Exp[I*Sqrt[v/k]*x] - Exp[-I*Sqrt[v/k]*x])/(2*I);

3. Modified expression: modifiedExpr = initialExpr[u0, x, t, k] /. eulerSubstitute

The output is: u0 (1 - Erf[x/(2 Sqrt[k t])])

My questions are:

1. Could someone please provide a detailed walkthrough of the mathematical steps involved in this transformation?
2. Specifically, how does the substitution and simplification using Euler's formula lead to the emergence of the error function in the final expression?
3. Are there any intermediate steps or mathematical properties that I might be missing or should be aware of to fully understand this transformation?

Any insights or detailed explanations would be greatly appreciated, as I am trying to deepen my understanding of these mathematical concepts.

Thank you in advance!