The verification of this Green's function solution is shown in Appendix A. The Green's function shows the Gaussian diffusion of the pointlike input with distance from the input (

We can use the Green's function
to write the solution for
in terms of
summing over its input values at points *z*'on the boundary at the initial time *t*'=0

Putting in the initial condtions at

To do the integral we change the variable to

The lower limit on the

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where substitution gives the dimensionless

In the first term we now complete the square to
get a new variable

The new lower limit on

After completing the square on the first term, the exponent simplifies to

Both integrals are now related to the Cumulative Distribution Function of the Normal Distribution

If we change

We now have our solution for the canonical

Finally, we use the facts that
, and that
, and the conversion

to get the Black-Scholes solution