Unknown letter ℑ used in an equation I need to write by hand the equation from the attached snapshot but I really don't know what letter is that seen in the front of square brackets [ . Can anyone help me ? 

 A: This is notation for the imaginary part of a complex number. It is a fraktur letter I, and its counterpart for the real part is a fraktur letter R. Thus, if $z=x+iy$ and $x,y$ are real, one writes
$$
\mathfrak{R}\,z=x\ \ \text{ and }\ \ \mathfrak{I}\,z=y.
$$
A good chart of the fraktur alphabet is in this Yale resource, which includes handwriting guidance, though some variation can be seen in google image searches. In LaTeX the quickest way to produce this is as the defaults of the commands \Re and \Im (though it is common for people to change this default), and which produce $\Re$ and $\Im$. Alternatively, you can use \mathfrak{R} and \mathfrak{I}, which produce $\mathfrak{R}$ and $\mathfrak{I}$; these are similar in the MathJax font displayed here but the specifics can vary on different systems. 
In my experience this is an older notation which has been superseded, pretty much everywhere, by the uppercase combinations $\mathrm{Re}$ and $\mathrm{Im}$:
$$
\operatorname{Re}(z)=x\ \ \text{ and }\ \ \operatorname{Im}(z)=y.
$$
These are typeset using \operatorname{Re} and \operatorname{Im}, although \mathrm usually works equally well. In handwritten work, most people denote the real and imaginary parts in this way; if you are making notes on a book which has the old notation, it is perfectly OK to use the new one, and it will not cause confusion for anyone reading your notes.
One thing to note is that fraktur is not a completely fixed font; it is more of a style of handwriting and there is a fair bit of variation in how each particular letter is drawn in different sources (for a sample, see the google images results for 'fraktur'). The particular $\mathfrak{I}$ in your image is relatively similar to how some fraktur fonts display the J (which sort of looks like $\mathfrak{J}$, but see the image results for the variation), but that is more of a fluke. The correct character to use is the I.
A: If it is actually the imaginary part of a complex variable, then just write $Im[\cdot]$ instrade of the curly character.
