Subtleties of the Exponential Function

I am a math enthusiast and I have always been fascinated by exponential functions. Most people know the definition or have a conceptual idea of what an exponential function is, but it is likely haven't seen some of the intricate and bizarre features of this seemly familiar function.

The simplest definition that is learned in school is that exponentials are repeated multiplication in the same way multiplication is repeated addition. This can be written as a^n=a*a*a(n times), where a^n denotes a to the power of b. Notice it is almost the exact same idea as a*b which is a*n=a+a+a...(n times). In general the idea of repeating an operation over and over again n times is called a hyperoperator. You can take the hyperoperator of power a^a^a^...(n times), but power are neither communitive nor associative, so their hyperoperator is not any where near as useful as the exponent. In the above definition because of the lack of associativity it is necessary to specify an order to the powers. Typically we use a^(a^(a^(...(n times)))... This definition of exponentiation is fairly straightforward to understand for example 2^3=2*2*2=8. Notice that the definition only makes sense when the power is a natural number i.e.{1,2,3...}. It is can be proven from this definition that (a^n)*(a^m)=a^(n+m), which follows simply from the fact that multiplying n times then m times is the same as multiplying n+m times. This is called the exponential addition property. The problem with this definition is that it only works when n is a natural number and we use powers all the time that are not for example if you type 2^3.75 in your calculator you will get an answer. The trick that is used to define powers for other numbers is called extension.

Extension is an idea that is used in math all the time. It is based on the idea that a definition may be limited to a certain context and we may want to "extend" it to a larger one. It is sort of like if someone starts selling hot dogs in New York and the hot dogs they sell are only popular in New York they might want to try to popularize them other places. In this case they might change their hotdog and add extra styles for other regions so that it is popular other places, but still has the same essential properties of the New York hotdog, and when people order it in New York it is still the same as it was before. For a concrete example imagine you are a biologist trying to come up with a definition of a mammal. Say that all you have is a list of animals you think should be mammals like people, dogs, elephants. The trick is that all of those produce milk for their young. In this case you could make this the definition of mammal. Now if you have an animal like a platypus that does other thing like lay eggs, you might be conflicted as to if this is a mammal of something else like a reptile, but because of the definition you made earlier, you can say with certainty that a platypus makes milk so it must be a mammal.

We can use extension on the exponential function. We say that the repeated multiplication definition only worked for natural numbers. In this case what we do it take a property that might work in general and demand it it be true in our new definition. The property we us is the exponential addition property (x^n)*(x^m)=x^(n+m). We are going to no longer require that n and m be natural thus we can rename them a and b. So (x^a)*(x*b)=x^(a+b), even though we have not defined what this means for a and b yet. There are two main ways of defining it. One is called a flow and is can be thought of as using trial and error. If we want x^0 for example we can substitute a=0 in to the above equation (x^0)*(x^b)=x(0+b)=x^b. The number that multiplies by something and doesn't change it for all of its values is 1. Therefor x^0=1. Likewise x^(-3) must satisfy (x^(-3))*(x^3)=x^(-3+3)=x^0=1. Therefor x^(-3) is the number that multiplies x^3 to give 1 which is 1/(x^3). For x^(1/2) it must satisfy (x^(1/2))*(x^(1/2))=x^(1/2+1/2)=x^1=x there for x^(1/2) must be the square root of x. In general there processes can give you x^(+-a/b) for b any natural number and a any natural number or zero. This gives the formula for all the rationales and the irrationals can be found using limits. This method of extension is called a flow, and it is not the one typically used in places like programming and analysis, but it is still useful an can be used sometimes when the next method fails or when the power is fairly simple.

There is another method of method of extension that can be used. Here we assume the reader is familiar with differential calculus. You can allow a^n to instead be a^x. Here we have not yet defined the exponential for non-natural x, but we still demand that (a^n)*(a^m)=a^(n+m) be satisfied even if n and m are allowed not to be natural. Then we take the derivative of a^x given by lim(h to 0) (a^(x+h)-a^x)/h. The most important part of this proof is that we use the exponential addition property to write a^(x+h)=(a^x)*(a^h). This is the step that allows us to extend the definition even though we have yet figured what these powers mean yet. Next we can factor this in to (a^x)*lim(h to 0) (a^h-1)/h=a^x*const. From here you can conclude that the derivative of an exponential is proportional to itself. By guessing a value E that makes this constant of proportionality 1 we can have d/dx(E^x)=E^x and we can use Taylor series to show E^x=1 +x+(1/2)x^2+(1/6)x^3+.... Plugging in x=1 we see that E=e which is one of the most enlightening examples in my opinion of where this mathematical constant comes from.

The Taylor Series definition also works in places where the flow definition doesn't. For example complex power we can use the Taylor series to define e^(ix)=cos(x)+isin(x). This is interesting for many reasons including providing relationship between exponentials and trigonometric functions that is extremely important in the study of linear differential equations and has applications in mechanics, electronics, quantum physics, ect. Note Trig Functions can also be written as exponentials using sin(x)=(e^(ix)-e^(-ix))/(2i) and cos(x)=(e^(ix)+e^(-ix))/2 This also gives us the polar form of a complex number and it can be proven every complex z can be written uniquely up to period as z=r*e^(iv). Typically the variable theta is used is place of were this article puts v. One interesting effect is the periodicity of the exponential on the imaginary line given by e^(x+2*i*pi*n))=e^x for n=1,2,3... This means that the exponential is not strictly one to one on the complex plane, and thus its inverse the logarithm will be multivalued. We know that the multivalued nature of many functions including the power function and the inverse trig function originates from the logarithm. For power functions a^b is defined as a^b=e^(b*ln(a)) a definition that is what calculators tend to use for implementing the power function. For complex numbers we can have multivaluedness by writing a=r*e^(iv) and then a^b=(r^b)*(e^(i*v*b)), but because all expressions with an added period are allowed this is (r^b)*(e^(i*v*b+2*pi*n*b)).

Given this it is easy to prove using basic number theory that a nonzero number to an integer will have 1 multiplicity, a nonzero to a rational number will have a multiplicity equal to the denominator of the rational number in lowest terms, and a nonzero number to an irrational power will have infinite multiplicity. A nonzero number to a complex number with a nontrivial imaginary component the multiplicity will tend to be infinite as well. The multiplicity means the number of values a function has. For example 16^(1/2)=4,-4 so its multiplicity is 2. This is a complex power which has a slightly different definition than a traditional power in that a traditional power is chosen to be single valued. Personally, I think we should have a different notation to distinguish between a real power and a complex power. Usually it is inferred from the context. Even as a complex power you may choose the single value type since you can use a branch cut which is typically used for multivalued functions to redefine them in a way that makes them singled valued.

The multivaluedness of the power function can be very confusing even for someone fairly familiar with it. For example one paradox is e^(1/2) should by the definition of complex power be +sqrt(e) and -sqrt(e). This contradicts what we are familiar with regarding the exponential function. The resolution in this paradox lies in the fact that the exponential we typically refer to as e^x or exp(x) is not actually e raised to the power of x in the sense of a complex power. This is where having separate notations would be helpful.

Another paradox is when you write a^b=e^(b*ln(a)). This is true for nearly all numbers and even other objects such as matrices. One place where it is not strictly true is 0^1 which would one the right hand side be written as e^(1*ln(0)), but of course ln(0) is undefined. The resolution here strictly speaking has to do with the fact that the definition does not apply when a=0, and the concept of extension must be used to say a^0=0 except when a=0 which gives 0^0 which truly is undefined. Using slightly looser mathematics if we interpret the right hand side as a limit then ln(0)=-infinity so e^(1*(-infinity))=e^(-infinity)=0 so the equation holds in a limiting sense.

We have seen that the power function although we tend to think of it as one function actually has a couple different definitions. For the most part these definitions yield the exact same results, but there are some special cases where the results differ. Among these definitions there are also many equivalent definitions that not only yield the same results, but are logically equivalent, but there around around three truly unique definitions of the exponential/power function.

Definition 1: Hyperoperator of multiplication: a^b means take a and multiply it by itself b times. Only valid when b is natural. Valid when a is any element of a group like structure that contains a basic multiplication operation.

Equivalent Definition 1: Recursive : a^b means the solution to the recursion relation a^b=a*a^(b-1) with initial condition a^0=1 except in the case a=0 in which case 0^b=0

Definition 2: Flow definition: a^b means when b is rational nonzero in the form +-m/n then multiply a by itself m times take n root and if negative take reciprocal. when b in irrational nonzero take limit of rational terms. When b is a^0=1 Valid when b is real and a is an element of any group like structure with a basic multiplication operation. Note that there are some cases this definition works even when the Taylor series doesn't for example functional powers. For readers who are interested look up either functional power or functional root. This definition is also typically used when a is a matrix and b is a simple power like 1/2, 1/3, ect.

Equivalent Definition 1: Extension of Exponential Addition Definition; Simply demand that the exponential addition property be satisfied for real numbers a.

Note: In the flow definition certain powers may or may not be defined, and may or may not be multivalued. In finite fields it is very common for a root to either not be defined of have multiple valued and the multiplicity is typically related to number theory. Also matrix roots tend to have infinite multiplicity. When dealing with zero type objects for example singular matrices, or groups with one element, even a^0 can have high multiplicity making it undefined. For example in the equation (x^a)*(x^b)=x^(a+b) if you substitute x=0, a=0 you get (0^0)*(0^b)=(0^b). 0^0 seems to be an multiplicative identity in this equation, and this provides motivation for a common convention of defining it to be 1, but notice if b is not equal to 0 then the equation becomes (0^0)*0=0. Normally the equation x*0=0 has solutions of x equaling the entire domain, the real numbers in this case suggesting the 0^0 has infinite multiplicity. Since this is just so weird we typically call 0^0 undefined and in general group theory you have to be careful using the power operation on zero like objects.

Definition 3 Taylor Series Definition: a^b means e^(b*ln(a)). e^x means 1+x+(1/2)x^2+(1/6)x^3+... Defining what ln(x) means is sort of tricky. For now lets say the definition is ln(x)=(x-1)-(1/2)(x-1)^2+(1/3)(x-1)^3+... This definition works when a is any real number between 0 and 2 including 2 and b and any group like object with a multiplicative structure. ln(x) can be extended to positive number many ways so this allows a to be any number greater than 0. This definition works in places the flow definition doesn't. We have b no longer being required to be a real number, and it works for b as many things including complex numbers, matrices, elements of finite fields, ect. This definition was derived by demanding that the exponential addition property be true. We can actually extend this again to values of b in which case the exponential addition property is no longer true. In general if two objects A,B don't commute that (x^A)*(x^B) won't equal x^(A+B). If A and B do commute then you can use the binomial theorem to prove the exponential addition property still works staring from the series definition. There are many ways to extend ln(a) to larger domains, and the most straightforward and general way is to simply defining it as the inverse of the exponential. In this case a can any thing that can be written as e^a=b in the domain being extended to. The series definition with the logarithm as the invers of the exponential is one of the most common definitions of exponentials. There are many different equivalent accepted definitions for logarithms.

Equivalent Definition 1: a^b means e^(b*ln(a)). e^x is the unique solution of dy/dx=y with initial condition y(0)=1. This definition uses any of the equivalent definitions for logarithms.

Equivalent Definition 2: a^b means e^(b*ln(a)). e^x is lim(n goes to infinity) (1+1/n*x)^(n*x). This definition uses any of the equivalent definitions for logarithms.

Equivalent Definition 3: a^b means e^(b*ln(a)). e^x is the inverse of the logarithm. This definition uses any of the equivalent definitions for logarithms, except as the inverse of an exponential which would be circular.

Here are some of the equivalent definitions of logarithms

Equivalent Definition 1: ln(x) means the inverse of the exponential. This definition uses any of the equivalent definitions of exponential except as the inverse of a logarithm which would be circular. The main advantage here is that the other definitions only extend the logarithm to positive numbers while this can extend it to any x such that there exists a y that satisfies e^y=x.

Equivalent Definition 2: ln(x) means the extension of the series (x-1)-(1/2)(x-1)^2+(1/6)(x-1)^3+... Here one way to extend the series is to use the property ln(a*b)=ln(a)+ln(b) and build up logarithms form numbers between 0 and 2. For example ln(3) can't be evaluated using the series, but ln(3)=ln(2)+ln(1.5) which can both be evaluated using the series.

Equivalent Definition 3: ln(x) means the integral from 1 to x of the function (1/x)

Definition 4: Complex Powers Definition: a^b means e^(b*ln(a)+b*2*pi*i*n) for n=1,2,3,... This is the only definition that actually yields different results of real positive number. Namely it is multivalued. This multivaluedness for reals can also be explained using the flow definition. In that case the answer to why does 4^(1/2) have two values is that there are two solutions to x^2=4. This is related to the fundamental theorem of algebra. Using the Taylor Series/Trigonometry definition this is explained by the fact that there are only two values for n before the sequence repeats. Note that there is also a complex logarithm. Ln(x) refers to the multivalued function and ln(x) refers to its branch cut which is single valued. For more on this look up complex logarithm.

So in conclusion there is possibly more to the exponential that one may have thought. The exponential is an extremely important function in mathematics and trying to think about all these definitions and get them all strait is good practice for becoming better at math an realizing some at the deep connections at the root of mathematics.