# Magic manipulation with the integrals

Today I have spend a lot of time to understand one thing. I do some stupid mistake, but can not see it at all. I hope somebody open my eye to my mistake.

I a littel bit simplify the problem, to make a idea much more clear.

We have function $$F=\int_0^\infty dT e^{i(m^2+ {\bf v}^2)T-\lambda T}$$ where $$m>0$$ and $$\lambda>0$$ and $$\bf v$$ is a two dimentional vector. $$\lambda$$ is a small reguralization parameter. I want to calculate $$\int d{\bf v} |F|^2$$. First way. We can calculate firstly the $$F=\frac{1}{\lambda-i(m^2+v^2)}$$ After that we can set $$\lambda=0$$ and perform the integration over $${\bf v}$$ $$\int d{\bf v} |F|^2=\pi\int_0^\infty d v^2 \frac{1}{(m^2+v^2)^2}=\frac{\pi}{m^2}$$ It is right result. The second way $$I=\int d{\bf v} |F|^2=\int d{\bf v} dT_1 dT_2 e^{i(m^2+ {\bf v}^2)(T_1-T_2)-\lambda (T_1+T_2)}$$ This integral is vell define. I can firstly perform the integration over $${\bf v}$$ $$I=i\pi\int \frac{dT_1 dT_2}{(T_1-T_2)} e^{i m^2(T_1-T_2)-\lambda (T_1+T_2)}$$ This integral looks like divergent in the $$T_1=T_2$$, but it converge becouse $$I$$ is a real. We do change of variables $$T_1=\xi(1+u)/2$$ and $$T_2=\xi(1-u)/2$$, we get $$I=i\pi\int_{-1}^1\frac{ du}{2u}\int_0^\infty d\xi e^{i m^2\xi u-\lambda\xi} =-\text{Im}\,\pi\int_0^1\frac{ du}{u}\int_0^\infty d\xi e^{i m^2\xi u-\lambda\xi}$$ Than, we can integrate over $$\xi$$: $$I=-\text{Im}\,\pi\int_0^1\frac{ du}{u}\frac{1}{\lambda-im^2u}=-\pi\int_0^1\frac{ du}{u}\frac{m^2 u}{\lambda^2+(m^2u)^2}=-\frac{\pi}{2}\int_{-1}^1\frac{m^2 du}{\lambda^2+(m^2u)^2}\\ =-\frac{\pi}{2\lambda}\int_{-m^2/\lambda}^{m^2/\lambda}\frac{dx}{1+x^2}=-\frac{\pi^2}{2\lambda}+\frac{\pi}{m^2}+O(\lambda)$$ We obtain the result whic contradict with the right one. The second result depends on regularization parameter $$\lambda$$. In the limit of $$\lambda\to 0$$ we obtain the $$I=-\infty$$. What I did wrong? How to do it correctly and reduce the first term $$\sim 1/\lambda$$?

In my real problem I can not do the first way because there is another functions in the exponent.

PLEASE, HELP ME. I DON'T UNDERSTAND WAT IS WRONG IN THIS CALCULATIONS!