What is the significance of the number e and Euler's formula?

[okay]

hi,
can anyone tell me what is the meaning of number e,i mean how it is discovered ?why derivative and antiderivative of this function same?i know it is very practical property of it but where did we get this number? and another thing eulers formula, which is
e^(i*pi)+1=0,
i also can't understand this equation. which comes from e^(i*pi)=sinx+i*cosx. where did we get this formula also?

 

[arildno]

hi,
can anyone tell me what is the meaning of number e,i mean how it is discovered ?

If I'm not mistaken, in the 17th century, banks became interested in the problem of how to compute continual (compound?) interest, and that pretty much drove the number "e" forth, since it is the limit of the expression $$(1+\frac{1}{n})^{n}$$ as n grows beyond bound.

$$why derivative and antiderivative of this function same?$$
I assume you are talking about the exponential, with "e" as its base.
Proofs of this is somewhat tricky.

i know it is very practical property of it but where did we get this number? and another thing eulers formula, which is
e^(i*pi)+1=0,
i also can't understand this equation. which comes from e^(i*pi)=sinx+i*cosx. where did we get this formula also?

It is not an equation, but an equality. It is a staggering result.

The complex exponential was originally shown to look like that by Euler, who used a power series representation of the exponential, and then saw how the trigonometric functions cropped up.

 

[naqo]

Okay, in order to understand a little more about the number e, one must analyze first
the limit presented above,$$\lim_{n\rightarrow \infty}(1+\frac{1}{n})^n$$. In particular,
one must prove that such limit exists, that means, that when you take n really big,
the number doesn't go to infinity. To do this, you may use the binomial theorem: in this
case, we obtain: $$(1+\frac{1}{n})^n = \sum_{k=0}^{n}\binom{n}{k}\frac{1}{n^k}$$.
But the binomial factor we can express it in a more convenient way and write it like this:
$$\sum_{k=0}^{n}\frac{1}{k!}n(n-1)...(n-k+1)\frac{1}{n^k}$$. Now, the number of those factors n(n-1)...(n-k+1) is k, so we can take the n^k and "share" it between all of them,

we obtain: $$\sum_{k=0}^{n}\frac{1}{k!}1(1-\frac{1}{n})...(1-\frac{1-k}{n})$$.

Let's call this sum S(n). Now, in orden to prove that the limit exists, whe have to prove that S(n+1)>S(n), this is, that the sequence of the S(n) is a increasing one, and then we may prove that it is smaller than a given number, so then, the limit must exist.

Following the same steps as before, we find that S(n+1) is equal to:
$$(1+\frac{1}{n+1})^{n+1}=\sum_{k=0}^{n+1}\frac{1}{k!}\frac{(n+1)n(n-1)...(n-k+2)}{(n+1)^k}$$
Now dividing each one of the factors in one of the n's from $$(n+1)^k$$ we get:
$$S(n+1)=\sum_{k=0}^{n+1}\frac{1}{k!}1(1-\frac{1}{n+1})...(1-\frac{k-1}{n+1})$$

Now if we compare S(n) and S(n+1), due to the fact that $$1-\frac{x}{n}$$ is smaller than $$1-\frac{x}{n+1}$$ (when x varies from 0 to k-1), we have found the proof that $$(1+\frac{1}{n})^{n}\leq(1+\frac{1}{n+1})^{n+1}$$

Now that we have proved that the sequence is monotone increasing, we must prove that there is a "top" or a "roof" (a limit) for the sum. We must see that beacuse all of the factors $$(1-\frac{1}{n})...(1-\frac{k-1}{n})$$ are smaller than 1, we note that if we replace all of them by 1, we would obtain something that is bigger than S(n), this is:
$$S(n)=\sum_{k=0}^{n}\frac{1}{k!}1(1-\frac{1}{n})...(1-\frac{1-k}{n})\leq\sum_{k=0}^{n}\frac{1}{k!}$$

So the question is really if $$\sum_{k=0}^{n}\frac{1}{k!}$$ is convergent or not. If it is convergent, then S(n) also is because we have proved that it is smaller.
Now we must consider the fact that $$k!\geq2^k$$ for k=0,1... so that means that: $$\frac{1}{k!}\leq\frac{1}{2^{k-1}}$$ and of course that if we take the sum of the smallers, it will be smaller than the sum of the largests:
$$\sum_{k=0}^{\infty}\frac{1}{k!}\leq\sum_{k=0}^{\infty}\frac{1}{2^{k-1}}$$.

Finally, we see that this last sum, $$\sum_{k=0}^{\infty}\frac{1}{2^{k-1}}=2+\sum_{k=1}^{\infty}\frac{1}{2^{k-1}}$$
and the LAST sum, $$\sum_{k=1}^{\infty}\frac{1}{2^{k-1}}=1+1/2+1/4+...=2$$ so that finally we obtain:

$$\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n\leq\sum_{k=0}^{\infty}\frac{1}{k!}\leq3$$
And obviously $$\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n\leq3$$

So, the number e is in fact finite and smaller than 3. There is no "analytical" way of actually finding the exact value of e, so you just ask a computer to calculate it.

I hope it will help you a bit okay, and sory about the english, i don't know it very well yet

 

[naqo]

aaaa sory about the symbols... just starting with latex...

 

[jostpuur]

aaaa sory about the symbols... just starting with latex...

You can edit your posts for 24h after the posting. Long posts with latex usually have typos, but it's not a problem because you can fix it quickly once you see how the post looks like.

 

[SiddharthM]

if you look at the taylor expansions of sinx, cosx, and e^x you get eulers equation when you replace x in the exponential with i*theta.

 

[jostpuur]

I put some similar calculations into this thread.

https://www.physicsforums.com/showthread.php?t=187750

can anyone tell me what is the meaning of number e,i mean how it is discovered ?why derivative and antiderivative of this function same?i know it is very practical property of it but where did we get this number?

Check in particular my post #8. There's one way to "find e".

and another thing eulers formula, which is
e^(i*pi)+1=0,
i also can't understand this equation. which comes from e^(i*pi)=sinx+i*cosx. where did we get this formula also?

It is not an equation, but an equality. It is a staggering result.

The complex exponential was originally shown to look like that by Euler, who used a power series representation of the exponential, and then saw how the trigonometric functions cropped up.

Some words of criticism:

Consider a following question. Let $f:[0,1]\to\mathbb{R}$ be $f(x)=1$ for all $x\in [0,1]$. Now, prove what the extension $\tilde{f}:[0,2]\to\mathbb{R}$ is.

Okey, that's a dumb question. There's no way of proving what the extension should be.

But then suppose we know what the exponential map $\mathbb{R}\to\mathbb{R}$, $x\mapsto e^x$ is. How do you prove that the extension of this mapping to the complex plane, $\mathbb{C}\to\mathbb{C}$, $z\mapsto e^z$, is something?

That is almost equally dumb question! You cannot prove what $e^z$ is, without some definition what it is supposed to be. Concepts such as analytic continuation or series are quite complicated concepts, so it wouldn't be unjustified to simply define

$$e^{x+iy} := e^x (\cos y + i\sin y)$$

This definition implies

$$e^{z_1} e^{z_2} = e^{z_1+z_2}$$

and agrees with the old real exponentiation when y=0, so the exponentiation notation is justified.

 

[rbj]

$$f(x) = e^x$$

is the inverse function to

$$g(x) = \log(x)$$

where $\log(x)$ is the "natural" logarithm and is defined to be this area or integral:

$$\log(x) = \int_{1}^{x} \frac{1}{u} du$$

note that it is a "1" in the numerator and not some other constant (which is what differentiates the natural logarithm from other bases of logarithm). we can see easily that

$$\log(ab) = \int_{1}^{ab} \frac{1}{u} du = \int_{1}^{a} \frac{1}{u} du + \int_{a}^{ab} \frac{1}{u} du = \int_{1}^{a} \frac{1}{u} du + \int_{1}^{b} \frac{1}{u} du = \log(a) + \log(b)$$

so it fits the main property of the logarithm and its inverse must be an exponential.

so e (a base of an exponential function) is that number so that

$$\log(e) = \int_{1}^{e} \frac{1}{u} du = 1$$

so the area under the 1/x curve from 1 to e is 1 unit of area.

a consequence of all of this is that, although it is the case that if

$$f(x) = A^x$$

then

$$f'(x) = \mathrm{(some constant)} \cdot A^x$$

that constant of proportionality is equal to 1 only if A = e = 2.71828182845905...

 

[Ben Niehoff]

where $\log(x)$ is the "natural" logarithm and is defined to be this area or integral:

$$\log(x) = \int_{1}^{x} \frac{1}{u} du$$

note that it is a "1" in the numerator and not some other constant (which is what differentiates the natural logarithm from other bases of logarithm).

Mostly right, except that having another constant instead of 1 will not change the base; it will merely add a constant:

$$\int_{C}^{x} \frac{1}{u} du = \int_{1}^{x} \frac{1}{u} du - \int_{1}^{C} \frac{1}{u} du = \ln x - \ln C$$

Edit: Oh, wait, you said "numerator". I thought you were referring to the other "1". Whoops.

 

[arildno]

I put some similar calculations into this thread.

https://www.physicsforums.com/showthread.php?t=187750



Check in particular my post #8. There's one way to "find e".





Some words of criticism:

Consider a following question. Let $f:[0,1]\to\mathbb{R}$ be $f(x)=1$ for all $x\in [0,1]$. Now, prove what the extension $\tilde{f}:[0,2]\to\mathbb{R}$ is.

Okey, that's a dumb question. There's no way of proving what the extension should be.

But then suppose we know what the exponential map $\mathbb{R}\to\mathbb{R}$, $x\mapsto e^x$ is. How do you prove that the extension of this mapping to the complex plane, $\mathbb{C}\to\mathbb{C}$, $z\mapsto e^z$, is something?

That is almost equally dumb question! You cannot prove what $e^z$ is, without some definition what it is supposed to be. Concepts such as analytic continuation or series are quite complicated concepts, so it wouldn't be unjustified to simply define

$$e^{x+iy} := e^x (\cos y + i\sin y)$$

This definition implies

$$e^{z_1} e^{z_2} = e^{z_1+z_2}$$

and agrees with the old real exponentiation when y=0, so the exponentiation notation is justified.

That is why I avoided the phrase that Euler "proved" the formula, he SAW how the trigonometric functions would crop up if it were possible to give some rigorous meaning to e^z for complex z.

 

[rbj]

BTW, Wikipedia has a pretty good page that explains how

$$e^z = e^{\mathrm{Re}\{z\} + i\mathrm{Im}\{z\}} = e^{\mathrm{Re}\{z\}} e^{i\mathrm{Im}\{z\}} = e^{\mathrm{Re}\{z\}} \left( \cos(\mathrm{Im}\{z\}) + i \sin(\mathrm{Im}\{z\}) \right) = e^{\mathrm{Re}\{z\}} \cos(\mathrm{Im}\{z\}) \ + \ i e^{\mathrm{Re}\{z\}} \sin(\mathrm{Im}\{z\})$$

i don't think we need to go through that here, do we?

 

[jostpuur]

BTW, Wikipedia has a pretty good page that explains how

$$e^z = e^{\mathrm{Re}\{z\} + i\mathrm{Im}\{z\}} = e^{\mathrm{Re}\{z\}} e^{i\mathrm{Im}\{z\}} = e^{\mathrm{Re}\{z\}} \left( \cos(\mathrm{Im}\{z\}) + i \sin(\mathrm{Im}\{z\}) \right) = e^{\mathrm{Re}\{z\}} \cos(\mathrm{Im}\{z\}) \ + \ i e^{\mathrm{Re}\{z\}} \sin(\mathrm{Im}\{z\})$$

i don't think we need to go through that here, do we?

Nowhere does it discuss the definition of the complex exponential function. It just talks about proving certain results. Not top quality explanation IMO.

 

[rbj]

BTW, Wikipedia has a pretty good page that explains how

$$e^z = e^{\mathrm{Re}\{z\} + i\mathrm{Im}\{z\}} = e^{\mathrm{Re}\{z\}} e^{i\mathrm{Im}\{z\}} = e^{\mathrm{Re}\{z\}} \left( \cos(\mathrm{Im}\{z\}) + i \sin(\mathrm{Im}\{z\}) \right) = e^{\mathrm{Re}\{z\}} \cos(\mathrm{Im}\{z\}) \ + \ i e^{\mathrm{Re}\{z\}} \sin(\mathrm{Im}\{z\})$$

i don't think we need to go through that here, do we?

Nowhere does it discuss the definition of the complex exponential function. It just talks about proving certain results. Not top quality explanation IMO.

i've heard this kind of stuff before and i have never gotten what the problem is.

jostpuur, do you have a pedagogical problem with the concept of adding an imaginary number to a real number? is such an addition defined? can we just treat the imaginary unit $i$ as a multiplicative factor?

when dealing with the definition of a complex exponential function, can we define it as the result of raising some number (real and positive for the time being) to a complex power? if addition is allowed regarding the real and imaginary numbers that make a complex number, and, if so, we can multiply such complex numbers together and apply the distributive property?

can we extend the property of exponentials regarding an exponent that is a sum of numbers to this addition of a real number to an imaginary number?

that's all we need to "assume" and the explanation in the Wiki article is just fine. all three proofs put in there cut the mustard just fine.

i don't get what could be bothering even the most anally-retentive rigorous mathematician here. whatever we define or assume about have real exponents to real numbers, we apply the same definitions, axioms, and rules to exponents that are a sum of a real and imaginary number. big deel.

 

[jostpuur]

i've heard this kind of stuff before and i have never gotten what the problem is.

jostpuur, do you have a pedagogical problem with the concept of adding an imaginary number to a real number? is such an addition defined? can we just treat the imaginary unit $i$ as a multiplicative factor?

No, definition of complex numbers isn't the problem, but instead the way how many introductions start proving properties of the complex exponential function without first discussing its definition. This easily gives the appearance, that it would be trivial what the exponential function is.

A good introduction would, for pedagogical reasons, first make it clear that the complex exponential function does not exists before it is defined. Then it should discuss what kind of properties we would like the complex exponential function to have. The exp(z1)*exp(z2)=exp(z1+z2) is the key property.

After this I don't care what the definition is. It can be given with the series, or with the real trigonometric functions, or perhaps in some other way that I don't know at the moment. Then it should be checked that the definition gives the desired properties for the exponential function, but the most important part is, that the definition at least exists.

when dealing with the definition of a complex exponential function, can we define it as the result of raising some number (real and positive for the time being) to a complex power?

No. The

$$z^n$$

is of course clear when $z\in\mathbb{C}$ and $n\in\mathbb{Z}$, but in general raising one complex number to the power of another goes like this

$$z_1^{z_2} := \exp(z_2\textrm{Log}(z_1))$$

and we already need the exponential function for this. Unless there exists some other way to define this, that I wasn't aware of.

(edit) Oh. hmmh.. you were talking about x^z when x is real and positive and z complex. But no.. the meaning of such power isn't obvious either.

 

[Ben Niehoff]

If you take two complex numbers z1 and z2 on the unit circle and multiply them, you will get a third complex number z3, also on the unit circle. Treating the complex plane as a 2D normed vector space, you can define the angle each number makes with the x-axis. What you will find is that

angle(z1) + angle(z2) = angle(z1 * z2)

What you'll notice is that our "angle" function translates products into sums. There is a natural isomorphism between this and the logarithm; the only question that remains is "Which logarithm?"

I'm not sure if there is a natural way to decide, besides the power series definition of exp(x). I suspect that there is, however. But I don't have time to look into it right now.

 

[rbj]

(edit) Oh. hmmh.. you were talking about x^z when x is real and positive and z complex. But no.. the meaning of such power isn't obvious either.

yes, that is what i was talking about. not only is x real and positive, it's a particular positive number, e, that has properties mentioned above that make it the base of the "natural" logarithm.

now this is obvious from the fundamental meaning of an exponential:

$$e^z = e^{\mathrm{Re}(z) + i \mathrm{Im}(z)} = e^{\mathrm{Re}(z)} e^{i \mathrm{Im}(z)}$$

don't you agree?

we can deal with multiplying by the real number $e^{\mathrm{Re}(z)}$ later (an operation with meaning that is obvious as long as we can derive a meaning for $e^{i \mathrm{Im}(z)}$, correct?

so then it is an issue of imparting meaning to

$$e^{i \mathrm{Im}(z)} = e^{i \theta}$$

where $\theta$ is real since we know, by axiom, that $\mathrm{Im}(z)$ is real.

at this point, the Wikipedia article does fine. it's plenty rigorous enough. either of the three proofs there completely answer the question of: "if $e^{i \theta}$ equals something, and if $i$ is a constant and has the property that when squared it's -1, what must that something be?".

what is missing?

 

[arildno]

It is not at all "obvious", jotspurr is right.

How you SHOULD proceed is, for example, to show that the complex function E(x)=cos(x)+isin(x)

satisfies the basic properties E(x+y)=E(x)E(y), E(0)=1, and so on.

 

[jostpuur]

now this is obvious from the fundamental meaning of an exponential:

$$e^z = e^{\mathrm{Re}(z) + i \mathrm{Im}(z)} = e^{\mathrm{Re}(z)} e^{i \mathrm{Im}(z)}$$

don't you agree?

If the exponential mappings

$$\mathbb{R}\to\mathbb{R}$$

and

$$\{iy\in\mathbb{C}\;|\;y\in\mathbb{R}\}\to\mathbb{C}$$

are already defined, then it makes sense to define

$$\exp:\mathbb{C}\to\mathbb{C}$$

with equation

$$\exp(x+iy)=\exp(x)\exp(iy).$$

I agree on this.

so then it is an issue of imparting meaning to

$$e^{i \mathrm{Im}(z)} = e^{i \theta}$$

where $\theta$ is real since we know, by axiom, that $\mathrm{Im}(z)$ is real.

at this point, the Wikipedia article does fine. it's plenty rigorous enough. either of the three proofs there completely answer the question of: "if $e^{i \theta}$ equals something, and if $i$ is a constant and has the property that when squared it's -1, what must that something be?".

what is missing?

It is impossible to prove what exp(iy) is, without a definition telling what it is supposed to be. If the proof does not use some definition, then it is not rigorous enough.

 

[rbj]

If the exponential mappings

$$\mathbb{R}\to\mathbb{R}$$

and

$$\{iy\in\mathbb{C}\;|\;y\in\mathbb{R}\}\to\mathbb{C}$$

are already defined, then it makes sense to define

$$\exp:\mathbb{C}\to\mathbb{C}$$

with equation

$$\exp(x+iy)=\exp(x)\exp(iy).$$

I agree on this.
It is impossible to prove what exp(iy) is, without a definition telling what it is supposed to be. If the proof does not use some definition, then it is not rigorous enough.

okay, so you agree that this

$$\exp(x+iy)=\exp(x)\exp(iy).$$

follows naturally. how about

$$\exp(i y_1 + i y_2)=\exp(i y_1)\exp(i y_2).$$

??

now we can define the value of e^iy to be the length of time, in milliseconds, of my dog's farts, but will that definition satisfy the existing properities of the exponential (such as the exponential of a sum is the product of exponentials)? if we treat the imaginary unit as a constant that it is (it sure as hell ain't a variable dependent on y), then there are existing rules for differentiating an exponential function that we can apply and then only "definition" for e^iy that fits those rules is e^iy = cos(y) + i sin(y). it's the only definition that fits. the Wikipedia article spells out 3 different ways that one might come upon that defintion. and then you can just verify that the definition makes sense by applying the fact above (along with the known trig indenties regarding the cosine and sine of a sum of two terms) .

so what's the big deal? i still don't get what your "missing link" is about.

 

[rbj]

It is not at all "obvious", jotspurr is right.

How you SHOULD proceed is, for example, to show that the complex function E(x)=cos(x)+isin(x)

satisfies the basic properties E(x+y)=E(x)E(y), E(0)=1, and so on.

sure, i agree. but how would you have guessed from no previous hints that

$$e^{i \theta} = \cos(\theta) + i \sin(\theta)$$

??

that is where those three proof in the Wikipedia article come in. (edit: the middle proof in the Wikipedia article already assumes you have a hint, a hypothesis, that $e^{i \theta} = \cos(\theta) + i \sin(\theta)$ ) and then proceeds to show that such a hypothesis satisfies the rules for differentiation for a function that is a ratio of the two equated expressions).

again, if $e^{i \theta}$ equals something (perhaps a real number, we find that $i^i$ is a real number, or maybe it's complex, or maybe it's a 10 dimensional vector, or some element in a Hilbert space, who know?, but it's assumed to be something), what must that "something" be in order for $e^{i \theta}$ to satisfy

$$e^{i (\theta_1 + \theta_2)} = e^{i \theta_1 + i \theta_2} = e^{i \theta_1} e^{i \theta_2} \quad \forall \theta_1,\theta_2$$

??

if $i$ is a constant, (and it axiomatically is not dependent on $\theta$), then this must be:

$$\frac{d}{d \theta} e^{i \theta} = i e^{i \theta}$$

and, remember, by axiom, you have $i^2 = -1$.

the rest is fiddling.

what else do you need (as far as definition)?

 

[jostpuur]

okay, so you agree that this

$$\exp(x+iy)=\exp(x)\exp(iy).$$

follows naturally.

In fact I'm not sure I agree, because I'm not sure what you mean by "follows naturally". From where does it follow? It could follow from our decision to define the left side with that equation. So initially the right side has a meaning, and the left side does not, and we define the left side with this equation. Can you see why this is not the same thing as proving what the left side is?

it's the only definition that fits. the Wikipedia article spells out 3 different ways that one might come upon that defintion.

But they don't talk about finding a reasonable definition. They simply show/derive/prove what the exponential function is, and let the reader believe that you can prove what the values of the exponential function is, without initially giving a clear definition to the function.

so what's the big deal?

The big deal is that "definition" and "proof" are different things, and they are not treated clearly in the popular introductions to complex exponential function. You are not treating them clearly either, since you first disagreed with me when I demanded definitions before proofs, but now you were instead already talking about "finding definitions" when still defending your own point.

 

[rbj]

In fact I'm not sure I agree, because I'm not sure what you mean by "follows naturally". From where does it follow?

that a complex number is defined axiomatically, to be the sum of a purely real number (such numbers are always real and non-negative when squared) and a purely imaginary number (such numbers are always real and non-postive when squared). that's an axiom, a definition of what it means to be a complex number. the place it follow is that the exponential of a sum is the product of exponentials.

It could follow from our decision to define the left side with that equation. So initially the right side has a meaning, and the left side does not, and we define the left side with this equation. Can you see why this is not the same thing as proving what the left side is?

the left side is an expression. we don't know yet fully what it means. it is a real and positve number (a particular number) raised to a complex power. we don't know if the result is real, complex, a tomato, or some other species of thing. but we know prior that the entire set of real numbers is a subset of complex numbers. then we hypothesize that it might be a complex number. then we ask, if e^iy is a complex number, what complex number must it be in order for the defining properties of an exponential function (of real numbers) to continue to hold? we make the hypothesis, see that we get an answer (some complex number with a real part and imaginary part) and then test the hypothesis against the existing properties of an exponential. if we find that

$$e^{iy} = \cos(y) + i \sin(y)$$

then we ask, could this be the only complex number that e^iy could be equal to, so then we test to see if

$$e^{iy} = \cos(y) + i \sin(y) + \epsilon$$

can satisfying the already accepted general properties of the exponential for any complex (or real) non-zero value of $\epsilon$. if it turns out that any non-zero value for $\epsilon$ shows that this deviated from those accepted properties, yet the zero value is consistent with the accepted properies of the exponential, then we know it has to be true (Euler's result, that is). at least for the case where we postulate that the result is complex.

But they don't talk about finding a reasonable definition.

baloney.

They simply show/derive/prove what the exponential function is, and let the reader believe that you can prove what the values of the exponential function is, without initially giving a clear definition to the function.

baloney.

the say clearly that whatever the exponential function is, it is such a function that satisfies the already existing properties of the exponential function known for real arguments. that is simply an extension of an existing definition.

The big deal is that "definition" and "proof" are different things,

i agree.

and they are not treated clearly in the popular introductions to complex exponential function. You are not treating them clearly either, since you first disagreed with me when I demanded definitions before proofs, but now you were instead already talking about "finding definitions" when still defending your own point.

that is pretty tangled speech to parse.

we say, ... oh hell, I'm not going to repeat it again.

 

[jostpuur]

The structure of the debate was pretty much this:

rbj: There's the proof for what exp is.

jostpuur: No you cannot prove it without some definition first.

rbj: You just look at the properties of exp, and prove what it is. I don't see a problem.

jostpuur: No you cannot prove it without the definition.

rbj: You just look at the properties, and then you just define it according to them, and then prove what it is. I still don't see what's the problem.

Okey there's no problem...

then we hypothesize that it might be a complex number. then we ask, if e^iy is a complex number, what complex number must it be in order for the defining properties of an exponential function (of real numbers) to continue to hold? we make the hypothesis, see that we get an answer (some complex number with a real part and imaginary part) and then test the hypothesis against the existing properties of an exponential

This seems to be respecting the principles of natural sciences very well, but that's not really how mainstream mathematics works.

Here's some counter examples to too unclear instructions for complex exponential.

----
Too little properties from real exponential:

We want exp(z1+z2)=exp(z1)*exp(z2).

For any fixed $\alpha\in\mathbb{R}$, define

$$\exp_{\alpha}(x+iy) = \exp(x)(\cos(\alpha y) + i\sin(\alpha y))$$

Now the desired property is satisfied with arbitrary alpha.
----

----
Too much properties from real exponential:

We want exp(z)>0, because also real exponential is positive.

We get this by choosing $\alpha = 0$, and we have $\exp_0(\mathbb{C})\subset\mathbb{R}$.
----


These counter examples prove, that a vague instruction "extend the properties of real exponential function to complex field" cannot be used to define a unique complex exponential function. The complex exponential function exists only after it has been defined with a clear definition, and not before.

 

[Ben Niehoff]

The missing piece of information is that we require $\exp(z)$ to be complex differentiable. That is, it should satisfy the Cauchy-Riemann equations:

$$f(x,y) = u(x,y) + i v(x,y)$$

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$

$$\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$$

----
Too little properties from real exponential:

We want exp(z1+z2)=exp(z1)*exp(z2).

For any fixed $\alpha\in\mathbb{R}$, define

$$\exp_{\alpha}(x+iy) = \exp(x)(\cos(\alpha y) + i\sin(\alpha y))$$

Now the desired property is satisfied with arbitrary alpha.

Writing

$$\exp_{\alpha} (x,y) = \exp(x) \cos \alpha y + i \exp(x) \sin \alpha y$$

we have

$$\frac{\partial u}{\partial x} = \exp(x) \cos \alpha y$$

and

$$\frac{\partial v}{\partial y} = \alpha \exp(x) \cos \alpha y$$

And so the Cauchy-Riemann equations are violated.

Too much properties from real exponential:

We want exp(z)>0, because also real exponential is positive.

We get this by choosing $\alpha = 0$, and we have $\exp_0(\mathbb{C})\subset\mathbb{R}$.

Choosing $\alpha = 0$, we have

$$\exp_0 (x,y) = \exp(x)$$

Then,

$$\frac{\partial u}{\partial x} = \exp(x)$$

and

$$\frac{\partial v}{\partial y} = 0$$

and so the Cauchy-Riemann equations are violated again.

If we demand that $\exp(z)$ be a complex differentiable function satisfying the properties of the exponential, then a unique function is specified, with $\alpha = 1$.

 

[Ben Niehoff]

Note that we could set

[tex]e^{x + iy} = e^{\alpha x} [\cos(\alpha y) + i \sin(\alpha y)]\qquad(1)[/itex]

but this is, in effect, a change of basis, allowing us to write

$$b^{x + iy} = b^x [\cos (y \ln b) + i \sin (y \ln b)]$$

where $b = e^{\alpha}$.

Also, note that (1) is inconsistent with the real exponential, if y=0. Therefore, again, the unique solution has $\alpha = 1$.

 

[Ben Niehoff]

Returning to the Cauchy-Riemann equations, we can consider the complex exponential to be a function

$$f(x,y) = u(x,y) + i v(x,y)$$

such that f, u, v, x, and y are real, and f satisfies

$$\frac{df}{dz} = f$$

where $z = x+iy$.

I don't have time to work this out right now, but this will lead to two coupled partial differential equations for u and v. Combined with the equations

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$

$$\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$$

we should be able to show that

$$u(x,y) = C \exp x \cos y$$

[tex]v(x,y) = C \exp x \sin y[/itex]

for arbitrary real constant C. Then we simply choose the solution that is consistent with $\exp x$ on the real line.

Hmm...no, I think C should turn out to be complex...but there is still a unique solution that is consistent with the real exponential.

 

[rbj]

The structure of the debate was pretty much this:

rbj: There's the proof for what exp is.

jostpuur: No you cannot prove it without some definition first.

rbj: You just look at the properties of exp, and prove what it is. I don't see a problem.

jostpuur: No you cannot prove it without the definition.

rbj: You just look at the properties, and then you just define it according to them, and then prove what it is. I still don't see what's the problem.

Okey there's no problem...
This seems to be respecting the principles of natural sciences very well, but that's not really how mainstream mathematics works.

Here's some counter examples to too unclear instructions for complex exponential.

----
Too little properties from real exponential:

We want exp(z1+z2)=exp(z1)*exp(z2).

For any fixed $\alpha\in\mathbb{R}$, define

$$\exp_{\alpha}(x+iy) = \exp(x)(\cos(\alpha y) + i\sin(\alpha y))$$

Now the desired property is satisfied with arbitrary alpha.
----

does it satisfy

$$e^{\beta z} = \big( e^{\beta} \big)^{z}$$

or

$$\frac{d}{dz} e^{\beta z} = \beta \ e^{\beta z}$$


for arbitrary but constant $\beta$ ?

----
Too much properties from real exponential:

We want exp(z)>0, because also real exponential is positive.

We get this by choosing $\alpha = 0$, and we have $\exp_0(\mathbb{C})\subset\mathbb{R}$.
----

so you make that assumption about other operations with complex numbers? we say that $i^2 < 0$ and we don't allow for that it the land of real numbers.

These counter examples prove, that a vague instruction "extend the properties of real exponential function to complex field" cannot be used to define a unique complex exponential function. The complex exponential function exists only after it has been defined with a clear definition, and not before.

i'm just saying that since complex numbers are the topic being that the argument of the exp function is a complex number, you do what you do for other extensions of functions of a real variable to having a possibly complex variable.

1. we require that the extension retain the "operational properties" of the function. for exp it's things like

$$e^{z_1+z_2} = e^{z_1}e^{z_2}$$


$$\big( e^{z_1} \big)^{z_2} = e^{z_1 z_2}$$and 2. we require that when the imaginary part of the argument is zero, it is consistent with the default definition that already exists for the natural expoential.

$$e^z = e^{\mathrm{Re}(z)}, \quad \mathrm{Im}\big\{ e^z \big\} = 0, \quad \mathrm{if} \ \mathrm{Im}\big\{ z \big\} = 0$$

i know that sometimes one can get surprized in that the species of animal you get back isn't always the same as the habitual assumption, like thinking that what one might get back from $\log(z)$ is a complex number when it's a vector of complex numbers, unless for example you make the further assumption of it returning a principle value.

you stretched what i did say to what i didn't.

the guy wanted to know why

$$e^{i \theta} = \cos(\theta) + i \sin(\theta)$$

for real theta.the Wikipedia article says that if we expect that those two rules of extensions above are implied, there is only one complex number that $e^{i \theta}$ can be equal to for the exponential to retain properties like its Maclauren series, or for it to appear in the denominator of a fraction with $\cos(\theta) + i \sin(\theta)$ in the numerator. apply the quotient rule (and remembering that $i^2 = -1$) and respecting a boundary condition that $e^{i 0} = 1$, and there is no other complex number that will fit the bill. to answer the guy's question, you don't have to make him/her worry about the possibility that $e^{i \theta}$ is a matrix, or an element of a Hilbert space.

 

[Ben Niehoff]

rbj, see my post above about the Cauchy-Riemann conditions. It is sufficient to demand that

$$e^{\alpha+\beta} = e^{\alpha}e^{\beta} \qquad \forall \alpha, \beta \in \mathbb{C}$$

$$e^z = e^{\Re(z)} \qquad \text{when } \Im(z) = 0$$

$$\frac{d}{dz}e^z \qquad \text{exists}$$

One can then prove that

$$\frac{d}{dz}e^z = e^z$$

and therefore $e^z$ is $C^{\infty}$.


Yet another angle of attack is to define $\ln z$ as the contour integral

$$\ln z = \int_{\gamma} \frac{d\zeta}{\zeta}$$

where $\gamma$ is some contour running from $1 + 0i$ to z. One can then define $e^z$ as the inverse of this function.

 

[Ben Niehoff]

So, here's an elaboration on the differential equation method I mentioned earlier. First, suppose we have a function

$$f(z) = f(z(x,y)) \qquad z(x,y) = x + iy$$

then

$$\frac{\partial f}{\partial x} = \frac{df}{dz}\frac{\partial z}{\partial x} = \frac{df}{dz} \qquad (1)$$

and

$$\frac{\partial f}{\partial y} = \frac{df}{dz}\frac{\partial z}{\partial y} = i \frac{df}{dz} \qquad (2)$$

Therefore, we see that

$$\frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y} = 0 \qquad (3)$$

This follows directly from the algebraic properties of $i$. Now, writing

$$f(x,y) = u(x,y) + i v(x,y)$$

we can re-write (3) as

$$\left( \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} \right) + i \left( \frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y} \right) = 0$$

$$\left( \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \right) + i \left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right) = 0 \qquad (4)$$

and these are the Cauchy-Riemann equations!


Now, what we want is to find some f(z) such that

$$\frac{df}{dz} = f$$

Using (1) and (2), we can therefore write:

$$\frac{\partial f}{\partial x} = f \qquad (5)$$

and

$$\frac{\partial f}{\partial y} = if \qquad (6)$$

So, again noting that $f(x,y) = u(x,y) + i v(x,y)$, we write (5) and (6) as

$$\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = u + iv$$

$$\frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y} = iu - v$$

These, then, yield four coupled first-order partial differential equations for u and v:

$$\frac{\partial u}{\partial x} = u \qquad (7)$$

$$\frac{\partial v}{\partial x} = v \qquad (8)$$

$$\frac{\partial u}{\partial y} = -v \qquad (9)$$

$$\frac{\partial v}{\partial y} = u \qquad (10)$$

(As an aside, the second two equations look like Hamilton's equations of motion, no?). Anyway, we can immediately integrate (7) and (8) as:

$$u = \eta_1(y) e^x \qquad \qquad v = \eta_2(y) e^x$$

where $\eta_1$ and $\eta_2$ are arbitrary functions of y alone. Plugging these into the second two equations, we get

$$\frac{\partial}{\partial y}(\eta_1(y) e^x) = -\eta_2(y) e^x$$

$$\frac{\partial}{\partial y}(\eta_2(y) e^x) = \eta_1(y) e^x$$

which simplify to

$$\frac{d\eta_1}{dy} = -\eta_2 \qquad (11)$$

$$\frac{d\eta_2}{dy} = \eta_1 \qquad (12)$$

Taking derivatives again with respect to y, we get

$$\frac{d^2\eta_1}{dy^2} = -\frac{d\eta_2}{dy}$$

$$\frac{d^2\eta_2}{dy^2} = \frac{d\eta_1}{dy}$$

Substituting back into (11) and (12), we decouple the equations as

$$\frac{d^2\eta_1}{dy^2} = -\eta_1$$

$$\frac{d^2\eta_2}{dy^2} = -\eta_2$$

These, then, can be integrated to yield

$$\eta_1(y) = A \cos y + B \sin y \qquad (13)$$

$$\eta_2(y) = C \cos y + D \sin y \qquad (14)$$

for arbitrary constants A, B, C, and D. Taking derivatives of (13) and (14) and plugging back into (11) and (12) yields

$$-A \sin y + B \cos y = -C \cos y + -D \sin y$$

$$-C \sin y + D \cos y = A \cos y + B \sin y$$

And so,

$$C = -B \qquad D = A$$

Therefore, we have

$$u(x,y) = \eta_1 e^x = e^x(A \cos y + B \sin y)$$

$$v(x,y) = \eta_2 e^x = e^x(-B \cos y + A \sin y)$$

And so

$$f(x,y) = u(x,y) + i v(x,y) = e^x(A \cos y + B \sin y) + i e^x(-B \cos y + A \sin y) \qquad (15)$$

for arbitrary constants A and B.

To choose a unique f(x,y), we require that

$$f(x,0) = e^x$$

Putting y=0 into (15), this yields

$$A e^x - iB e^x = e^x$$

and so, A=1 and B=0. Therefore, the unique extension of $\exp z$ over the complex numbers is

$$\exp z = \exp (x + iy) = e^x \cos y + i\, e^x \sin y$$

This is the only function over the complex numbers which satisfies both

$$\frac{d}{dz} \exp z = \exp z$$

and

$$\exp (x + i0) = e^x$$

Q.E.D.

 

[rbj]

rbj, see my post above about the Cauchy-Riemann conditions. It is sufficient to demand that

$$e^{\alpha+\beta} = e^{\alpha}e^{\beta} \qquad \forall \alpha, \beta \in \mathbb{C}$$

$$e^z = e^{\Re(z)} \qquad \text{when } \Im(z) = 0$$

$$\frac{d}{dz}e^z \qquad \text{exists}$$

i agree with you, Ben. i don't think we just define the complex exponential function as

$$e^z = e^{x+iy} = e^x \cos(y) + i e^x \sin(y)$$

i think we (as did Euler) set out to find a "definition" (explicit functions for the real and imaginary parts of e^z) so that the behavior of the expontial function is carried over to the world of complex numbers and that when the imaginary part of the argument is zero, that the exponential behaves as it previously had with real input.

 

[Ben Niehoff]

Right. So, after some investigation, I have found that there are two looser definitions one might use:

Definition 1:
$\exp(z)$ is the unique function over $\mathbb{C}$ such that:

1. $\frac{d}{dz}(\exp(z))$ exists, and

2. $\exp(x+i0) = e^x$ for all real numbers x.

Definition 2
$\exp(z)$ is the unique function over $\mathbb{C}$ such that:

1. $\frac{d}{dz}(\exp(z)) = \exp(z)$, and

2. $\exp(0) = 1$.

Either definition is sufficient, and they are equivalent.

 

[jacobrhcp]

I was bored in class so I did this, don't really know if it's any use here, but I think so and found it very interesting:

d/dx[exp(x)]=lim h->0 $$\frac{exp(x+h)-exp(x)}{h}$$ = exp(x) lim h->0 $$\frac{exp(h)-1}{h}$$=exp(x) lim h->0 $$\frac{\sum\frac{h^k}{k!}-1}{h}$$, with the sum from 0 to infinity,

= exp(x) lim h->0 $$\frac{\sum\frac{h^k}{k!}}{h}$$, with the sum from 1 to infinity

= exp(x) lim h->0 $$\sum\frac{h^{(k-1)}}{k!}$$

and all terms in the sum go to zero, except the first, which goes to 1... so:

= exp(x)

so, when you define exp(x) by $$\sum\frac{x^k}{k!}$$, the deravative of exp(x) is exp(x)

 

[Gib Z]

Yes, and that result would have been much easier if you differentiated the series directly, term by term. Or even if you used the limit definition, at the point $$\lim_{h\to 0} \frac{e^h -1}{h}$$ you could have replaced with the series definition there, subtract one from it and then divide by h, you have the same series again.

 

[jacobrhcp]

well that's what I did >_>, at least the second thing.

 

[Gib Z]

My bad, I didn't pay attention to the intervals of summation.