In my book, the defination of Fourier transform is $$F(\lambda)=\int_{-\infty}^{+\infty}f(t)e^{i\lambda t}dt$$ While the reverse one is: $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}F(\lambda)e^{-i\lambda t}dt$$ But in other place (here as well), I always encounter another "just on the contrary" system like: $$F(\lambda)=\int_{-\infty}^{+\infty}f(t)e^{-i\lambda t}dt$$ You can see that the sign of exponential part is changed.
Why will this happen?
As long as you don't consider the inverse transformation it is unimportant whether you write $e^{i\lambda t}$ or $e^{-i\lambda t}$ in the definition of $F$.
As soon as you look at $F$ and $F^{-1}$ at the same time you have to put a minus sign either in the definition of $F$ or in the resulting $F^{-1}$. This is so because the Fourier transform is sort of a "$90^\circ$ rotation, or multiplication by $i$, in function space", and $(i)\cdot(i)\ne 1$, but $(i)\cdot(-i)=1$.
This is a problem I also faced as a student. This happens everywhere in mathematics and perhaps in other subjects too. It is better to use different names when you have a different meaning or a different formula. It is better to use one standard than every one use their own formula and use the same name.
Matrix canonical forms also different mathematicians use slightly different forms. When a student reads both the forms, he is not sure which one to use in the exam.
In abstract algebra too, groups and rings and their properties are slightly different from different authors.
