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Lost the plot at "... and for sensible v, this is equivalent to ..." :(
There's certain forms like that that have well known values that they converge to as you continue adding terms into infinity. Sometimes that convergence is only possible if your domain is limited, eg. [0,1].
Lost the plot at "... and for sensible v, this is equivalent to ..." :(
For that, I would look at early calculus/pre-calc, I think, examining infinite series and their properties and equivalencies. There's certain forms like that that have well known values that they converge to as you continue adding terms into infinity. Sometimes that convergence is only possible if your domain is limited, eg. [0,1].
For those interested in looking slightly more into the characteristic function, it may be worth pointing out that the characteristic function is equal to the Fourier-transform (with the sign of the argument being reversed) of the probability distribution in question. In my own experience teaching teaching probability theory to physicists and engineers, establishing this connection is often a good way of helping peopl…
I mean, it was useful for me to think about like a translation between sets and logic (this variable x is in the set xor not) into functions (a function f(x) that returns 1 or true whenever x is in set S)
how the heck is that a fourier transform!??
For those interested in looking slightly more into the characteristic function, it may be worth pointing out that the characteristic function is equal to the Fourier-transform (with the sign of the argument being reversed) of the probability distribution in question. In my own experience teaching teaching probability theory to physicists and engineers, establishing this connection is often a good way of helping peopl…
but isn't a characteristic function just "the" way to bridge the gap between sets, functions, and logic(? ...a 3way bridge!?) I mean, it was useful for me to think about like a translation between sets and logic (this variable x is in the set xor not) into functions (a function f(x) that returns 1 or true whenever x is in set S) how the heck is that a fourier transform!??
Earlier quoted context omitted.
For that, I would look at early calculus/pre-calc, I think, examining infinite series and their properties and equivalencies. There's certain forms like that that have well known values that they converge to as you continue adding terms into infinity. Sometimes that convergence is only possible if your domain is limited, eg. [0,1].
That clears it, thanks! I didn't figure out that "sensible" referred to convergent G(t).
Earlier quoted context omitted.
For that, I would look at early calculus/pre-calc, I think, examining infinite series and their properties and equivalencies. There's certain forms like that that have well known values that they converge to as you continue adding terms into infinity. Sometimes that convergence is only possible if your domain is limited, eg. [0,1].
That clears it, thanks! I didn't figure out that "sensible" referred to convergent G(t).