This Quantum World/Appendix/Exponential function

The exponential functionEdit

We define the function   by requiring that


The value of this function is everywhere equal to its slope. Differentiating the first defining equation repeatedly we find that


The second defining equation now tells us that   for all   The result is a particularly simple Taylor series:


Let us check that a well-behaved function satisfies the equation


if and only if


We will do this by expanding the  's in powers of   and   and compare coefficents. We have


and using the binomial expansion


we also have that



The function   obviously satisfies   and hence  

So does the function  

Moreover,   implies  

We gather from this

  • that the functions satisfying   form a one-parameter family, the parameter being the real number   and
  • that the one-parameter family of functions   satisfies  , the parameter being the real number  

But   also defines a one-parameter family of functions that satisfies  , the parameter being the positive number  

Conclusion: for every real number   there is a positive number   (and vice versa) such that  

One of the most important numbers is   defined as the number   for which   that is:  :


The natural logarithm   is defined as the inverse of   so   Show that


Hint: differentiate