Let n be a non-negative integer. In the sense of definition 4 and by induction, . Therefore Using Taylor series, This shows that definition 4 implies definition 2. In the sense of definition 2, Besides, This shows that definition 2 implies definition 4. Characterization 2 ⇒ characterization 6 In the sense of definition 2, Characterization 3 ⇔ characterization 4 Characterisation 3 involves defining the natural logarithm before the exponential function is defined. First, This means that the natural logarithm of equals the (signed) area under the graph of between and . If , then this area is taken to be negative. Then, is defined as the inverse of , meaning that by the definition of an inverse function. If is a positive real number then is defined as . Finally, is defined as the number such that . It can then be shown that : By the fundamental theorem of calculus, the derivative of . We are now in a position to prove that , satisfying the first part of the initial value problem given in characterisation 4: Then, we merely have to note that , and we are done. Of course, it is much easier to show that characterisation 4 implies characterisation 3. If is the unique function satisfying , and , then can be defined as its inverse. The derivative of can be found in the following way: If we differentiate both sides with respect to , we get Therefore, Characterization 5 ⇒ characterization 4 The conditions {{math|1=f'''(0) = 1}} and imply both conditions in characterization 4. Indeed, one gets the initial condition by dividing both sides of the equation by , and the condition that follows from the condition that and the definition of the derivative as follows: Characterization 6 ⇒ characterization 4 In the sense of definition 6, By the way , therefore definition 6 implies definition 4. References Walter Rudin, Principles of Mathematical Analysis, 3rd edition (McGraw–Hill, 1976), chapter 8. Edwin Hewitt and Karl Stromberg, Real and Abstract Analysis'' (Springer, 1965). Mathematical analysis Exponentials Exponential function Articles containing proofs