Where does the exponential function come from in roots?

[TachyonLord]

For example, in linear differential equations, there might be these questions where we'd directly use e^∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?

 

[fresh_42]

For example, in linear differential equations, there might be these questions where we'd directly use e^∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?

That's a good question. I like to see it this way: The exponential function solves a fix point problem, namely $y'=y$. A general differential equation is a kind of variation of it, $y'=f(y)$, so it's plausible that a some sort of modification of the exponential function, $c \cdot e^{g(x)}$, comes into play.

Another point of view is the fundamental (and defining!) formula $e^x\cdot e^y=e^{x+y}\,.$ It translates the multiplicative structure of some curved phase space, which the solutions to a differential equation system are, into the linearity of its tangent spaces, or for short $e^0=1$ or $e^{\operatorname{trace}}=\det$. In a differential equation system we have some statements about tangent spaces, i.e. something linear: $x+y$, and we are searching for the curved solution to which these are the tangent spaces, i.e. $x+y \mapsto f(x)\cdot f(y)\,.$ And such a relation defines the exponential function. The next prominent functions are periodic ones, the trigonometric functions. However, a little detour to the complex numbers shows us, that they are, too, a combination of exponential functions.
So for me it is the property $e^{\text{linear}} = \text{curved}\; , \;e^{\text{tangents}}=\text{flow}\,.$

 

[TachyonLord]

That's a good question. I like to see it this way: The exponential function solves a fix point problem, namely $y'=y$. A general differential equation is a kind of variation of it, $y'=f(y)$, so it's plausible that a some sort of modification of the exponential function, $c \cdot e^{g(x)}$, comes into play.

Another point of view is the fundamental (and defining!) formula $e^x\cdot e^y=e^{x+y}\,.$ It translates the multiplicative structure of some curved phase space, which the solutions to a differential equation system are, into the linearity of its tangent spaces, or for short $e^0=1$ or $e^{\operatorname{trace}}=\det$. In a differential equation system we have some statements about tangent spaces, i.e. something linear: $x+y$, and we are searching for the curved solution to which these are the tangent spaces, i.e. $x+y \mapsto f(x)\cdot f(y)\,.$ And such a relation defines the exponential function. The next prominent functions are periodic ones, the trigonometric functions. However, a little detour to the complex numbers shows us, that they are, too, a combination of exponential functions.
So for me it is the property $e^{\text{linear}} = \text{curved}\; , \;e^{\text{tangents}}=\text{flow}\,.$

Thank you so much :)