This Quantum World/Appendix/Exponential function

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The exponential function[edit]

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