09/16/2021, 11:13 PM

- quote -

As a small example :

integral from 1 to +oo [ t^x g(t) dt ]

with g(t) = exp(- ln(t)^2 )

equals :

(1/2) * ( erf((x+1)/2) +1) * sqrt(pi) * exp( (1/4)* (x+1)^2 ).

I find this fascinating.

- end quote -

another useful example is

integral from 1 to +oo [ t^x g(t) dt ]

with g(t) = ln(t)^v

equals :

v! (x-1)^(-v-1)

for Re(x)>1.

thereby connecting to laurent series and more.

This might be well known but for completeness and relevance I had to add it.

regards

tommy1729

As a small example :

integral from 1 to +oo [ t^x g(t) dt ]

with g(t) = exp(- ln(t)^2 )

equals :

(1/2) * ( erf((x+1)/2) +1) * sqrt(pi) * exp( (1/4)* (x+1)^2 ).

I find this fascinating.

- end quote -

another useful example is

integral from 1 to +oo [ t^x g(t) dt ]

with g(t) = ln(t)^v

equals :

v! (x-1)^(-v-1)

for Re(x)>1.

thereby connecting to laurent series and more.

This might be well known but for completeness and relevance I had to add it.

regards

tommy1729