The upper bound of an integral involving exponential function

Question

Thanks for your attention, I need to obtain the upper bound of the following integral, $$I = \int_0^\infty {{e^{ - {{\left( {\ln x + a} \right)}^2}}}} \cdot {e^{ - {{\left( {\ln \left( {x + b} \right) + c} \right)}^2}}}d\left( {\ln x} \right)$$ where $a$, $b$, $c$ are constant and $a$<0, $b \geqslant 0$. Thanks a lot.

Answer 1

It is not a general solution, but it may point toward one. We can start considering $b=0$. Then you have $$ I = \int_0^\infty {{e^{ - {{\left( {\ln x + a} \right)}^2}}}} \cdot {e^{ - {{\left( {\ln {x } + c} \right)}^2}}}d\left( {\ln x} \right) $$ And you can make the substitution $y=\ln x$ therefore changing the limit of the integral to $-\infty$ and $+\infty$ getting you the integral to study $$ I = \int_{-\infty}^\infty {{e^{ - {{\left( {y + a} \right)}^2}}}} \cdot {e^{ - {{\left( {y + c} \right)}^2}}}d {y} $$ that should be easy to study. I will expand this answer as soon as I have done more, but maybe it will help the OP in some way (I hope). If posting such an answer is considered bad form (because it not really an answer) just drop a comment below and I will delete it.

And to approach this problem you can find a nice work here: http://www.tina-vision.net/docs/memos/2003-003.pdf

EDIT: an additional information that could help you is that we dealing here with a convolution of functions, something that has quite a solid theory behind it... You may want to look into it.