This Quantum World/Appendix/Exponential function

The exponential functionEdit

We define the function   by requiring that

   and   

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

 

Voilà.

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