How did YOU get that value for z?

I'm not aware of any way to do it in closed form.

There is a closed form solution involving the LambertW function: https://en.wikipedia.org/wiki/Lambert_W_function

Here are 20 digits worth of the solution:

 

There is a closed form solution involving the LambertW function: https://en.wikipedia.org/wiki/Lambert_W_function

Here are 20 digits worth of the solution:

View attachment 172794

I'm pretty sure we can write almost anything in closed form, we just have to give a name to a suitable power series equation or other such critter and then use the name.

The choice of which functions and operators are included in the set of allowable ones is certainly arbitrary, but I'd be willing to guess that the LambertW function is seldom included.

 

I'm pretty sure we can write almost anything in closed form, we just have to give a name to a suitable power series equation or other such critter and then use the name.

The phrase "closed form" is not generally used in the way you are describing. You make it almost trivial. Who is the "we" you are referring to that simply gives a name to a suitable power series and then uses the name, claiming it's a "closed form"?

The choice of which functions and operators are included in the set of allowable ones is certainly arbitrary

The choice is not arbitrary; the functions and operators included must be "well known". See: https://en.wikipedia.org/wiki/Closed-form_expression

One might ask "well known" by whom? To me it means "well known" by the mathematical and scientific communities at large.

but I'd be willing to guess that the LambertW function is seldom included.

The LambertW function is especially suited for use in the mathematics of circuits involving a resistance in series with a semiconductor junction. An internet search will find many examples. Common math software supports it, such as Matlab, Mathematica, etc.

 

The phrase "closed form" is not generally used in the way you are describing. You make it almost trivial. Who is the "we" you are referring to that simply gives a name to a suitable power series and then uses the name, claiming it's a "closed form"?

I just did a Google search for "closed form solution" and the first hit I got was this:

http://mathworld.wolfram.com/Closed-FormSolution.html

"An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum."

How is that SO different from what I said?

 

Hello,

Well to me if it includes an infinite series then it's not really closed form although i think it is considered analytic.
If we could include an infinite series then we'd have nearly everything being in closed form.
Maybe we should look at what is not considered closed form and compare.
I dont usually consider any numerical approach as being in closed form. When a numerical approach is the only way to solve an equation, i tend to think it is not in closed form because then we can not use the result in the next stage of a complete calculation and have it be in a relatively simple form. A symbolic example:
y=2*x
vs
y=f(x)

where f(x) can only be solved numerically. Then the next stage could be:
z=g(y)

and now if f(x) was in closed form, then we could easily write the last funciton:
z=g(f(x))

but because f(x) has to be solved numerically, even if g(x) is simpler it becomes a problem.
Consider:
y=f(x,y)

and we see the problem now. That expression, assuming y can not be solved for explicitly, makes the next stage complicated to calculate:
z=g(y=f(x,y))

See how strange that gets? That's because y cant be solved for explicitly and therefore there is no symbolic solution.

I guess we should put more thought into this though.

 

Hello,

Well to me if it includes an infinite series then it's not really closed form although i think it is considered analytic.

So do you consider things like

y = ln(x)
a = cos(b)
d = Atan(theta)

to be not closed form?

 

So do you consider things like

y = ln(x)
a = cos(b)
d = Atan(theta)

to be not closed form?

Hi,

Yeah i see what you are getting at i think.
But as i said, more thought, more thought ;-)

 

Hi,

Yeah i see what you are getting at i think.
But as i said, more thought, more thought ;-)

Hi again,

I think we might divide the functions into at least two classes, those that are strictly defined as closed and those that are not so strictly defined.
For example elementary functions such as,
y=cos(x)

would be strictly defined as closed, while a power series would be not so strictly defined and subject to interpretation.

Also subject i think is an implicit expression such as:
y=x*cos(y)

since we dont know what y is analytically it is hard to call that closed.
On the other hand, if we can find a solution to that then we can resolve this into closed form.

I guess what i am trying to get at is that some functions can not be calculated without resorting to numerical solutions because they can not reduce to a set of elementary functions. I've run into this a lot and it presents a problem to the next phase of calculations because you can never solve for the required variable without using numerical solving.

Again more thought

 

Hi again,

I think we might divide the functions into at least two classes, those that are strictly defined as closed and those that are not so strictly defined.
For example elementary functions such as,
y=cos(x)

would be strictly defined as closed, while a power series would be not so strictly defined and subject to interpretation.

First, what is an "elementary function"? Sounds an awful lot like you are talking about a function from a "generally-accepted set". But how do you determine which functions are "elementary" and which ones aren't?

Furthermore, cos(x) IS a power series!

How is writing it as

\(y \; = \; \cos(x)\)

somehow "strictly defined" but writing it as

\( \sum_{k=0}^{/infinity} {\(-1\)}^k \frac{x^{2k}}{\(2k\)!}\)

somehow not so strictly defined and subject to interpretation?

Also subject i think is an implicit expression such as:
y=x*cos(y)

since we dont know what y is analytically it is hard to call that closed.
On the other hand, if we can find a solution to that then we can resolve this into closed form.

Simple.

fred(y) = y/cos(y) (plus definitions at points where cos(y) = 0)

y = arcfred(x)

There. Closed form.

All we need to do is get enough people to adopt our function fred() in order for it to be an "elementary function" or, as TheElectrician would have it, "well-known".

Heck, it probably already is by some other name. Think of the sinc() function. Perhaps there's already a cosc() function?

I guess what i am trying to get at is that some functions can not be calculated without resorting to numerical solutions because they can not reduce to a set of elementary functions. I've run into this a lot and it presents a problem to the next phase of calculations because you can never solve for the required variable without using numerical solving.

Okay, so let's go back to

y = cos(x)

which you have no problem calling an elementary function. So how would you find the value of y if the value of x is 2.57 without resorting to numerical solutions?

 

First, what is an "elementary function"? Sounds an awful lot like you are talking about a function from a "generally-accepted set". But how do you determine which functions are "elementary" and which ones aren't?

Furthermore, cos(x) IS a power series!

How is writing it as

\(y \; = \; \cos(x)\)

somehow "strictly defined" but writing it as

\( \sum_{k=0}^{\infty} {\(-1\)}^k \frac{x^{2k}}{\(2k\)!}\)

somehow not so strictly defined and subject to interpretation?



Simple.

fred(y) = y/cos(y) (plus definitions at points where cos(y) = 0)

y = arcfred(x)

There. Closed form.

All we need to do is get enough people to adopt our function fred() in order for it to be an "elementary function" or, as TheElectrician would have it, "well-known".

Heck, it probably already is by some other name. Think of the sinc() function. Perhaps there's already a cosc() function?



Okay, so let's go back to

y = cos(x)

which you have no problem calling an elementary function. So how would you find the value of y if the value of x is 2.57 without resorting to numerical solutions?

Hi,

Well there is a difference between:
y=f(x)
and
y=f(x,y)

as the second is implicit and so we cant solve for y. Yes, maybe for some functions we can, but for many we can not.
So closed form to me is when we can actually solve for the variable of interest symbolically.

The example of cos(x) and the infinite sum is not a good one i think because the infinite sum would be reduced to cos(x) in order to put it into closed form.

Maybe it is always subject to interpretation? Just like what we call 'linear' is subject to a judgement based on the independent variable.

Maybe a limited power series is a closed form but an infinite series cant be. This seems like a good way to look at it because:
y=1+x+x^2+x^3
is closed but then
y=1+x+x^2+x^3+...+x^n
is hard to call closed.

Let me reiterate the problem i've seen again.
We have a function:
y=f(x)

and it is easy to use in another function:
z=g(f(x))

but if we have:
y=f(x,y)

we cant put it into that same function g() because we would end up with:
z=g(f(x,y))
which can never be solved because we dont know what y is, unless as before we can actually solve for it, but when we cant not that is when the problem comes up certainly not when we can.

Take this function:
x^y=e^y+e^(-y)

We can solve this for x but can we solve this for y? Maybe, but let's say we cant. Then how do we calculate:
z=2*y+3

First, we are implying that we end up with:
z=2*f(x)+3

but we dont know that the function f(x) is. To solve, we have to know what x is first, then solve the first equation for y, then insert that numerical value into the expression for z.

Note that the function you substituted for mine is able to be solved so that's not the same thing.

This kind of problem pops up a lot. One i remember best is the exact solution to the full wave (or half wave) rectifier with filter cap and resistive load with maybe a series resistance too and solving based on the voltage responses. The expression that comes up can only be solved numerically. Can it still be said to be in closed form?

Maybe it would be better to just abandon that classification because it is sometimes too subject to interpretation.

 

Hi,

Well there is a difference between:
y=f(x)
and
y=f(x,y)

as the second is implicit and so we cant solve for y. Yes, maybe for some functions we can, but for many we can not.
So closed form to me is when we can actually solve for the variable of interest symbolically.

You suggested the following:

y = x·cos(y)

I solved it for y symbolically by defining a function fred() and it's inverse.

There is fundamentally NO difference between using fred() to solve it symbolically than using cos(), ln(), exp(), or any other power series function and/or their inverses.

The example of cos(x) and the infinite sum is not a good one i think because the infinite sum would be reduced to cos(x) in order to put it into closed form.

How is reducing the infinite sum for cos(x) to cos(x) any different than reducing the infinite sum for fred(x) to fred(x) or the infinite sum for sinc(x) to sinc(x) or the infinite sum for ln(x) to ln(x)?.

Fundamentally, the only difference is that cos(x) is a name for a power series function that essentially everyone agrees is a "well-known" or "generally-accepted" function for the purposes of deciding which functions can appear in closed-form solutions, which fred(x) likely isn't. But, as Wolfram states explicitly, this is an inherently arbitrary decision.

Maybe it is always subject to interpretation?

Which is the major point of this side discussion -- which functions and which operations are considered "generally-accepted" or "well-known"? Who decides? The Wolfram entry explicitly points out that there is no universal agreement and that different branches of mathematics have different sets.

Maybe a limited power series is a closed form but an infinite series cant be. This seems like a good way to look at it because:
y=1+x+x^2+x^3
is closed but then
y=1+x+x^2+x^3+...+x^n
is hard to call closed.

Let me reiterate the problem i've seen again.
We have a function:
y=f(x)

and it is easy to use in another function:
z=g(f(x))

but if we have:
y=f(x,y)

we cant put it into that same function g() because we would end up with:
z=g(f(x,y))
which can never be solved because we dont know what y is, unless as before we can actually solve for it, but when we cant not that is when the problem comes up certainly not when we can.

Take this function:
x^y=e^y+e^(-y)

We can solve this for x but can we solve this for y? Maybe, but let's say we cant. Then how do we calculate:
z=2*y+3

First, we are implying that we end up with:
z=2*f(x)+3

but we dont know that the function f(x) is. To solve, we have to know what x is first, then solve the first equation for y, then insert that numerical value into the expression for z.

Note that the function you substituted for mine is able to be solved so that's not the same thing.

This kind of problem pops up a lot. One i remember best is the exact solution to the full wave (or half wave) rectifier with filter cap and resistive load with maybe a series resistance too and solving based on the voltage responses. The expression that comes up can only be solved numerically. Can it still be said to be in closed form?

Maybe it would be better to just abandon that classification because it is sometimes too subject to interpretation.[/QUOTE]

It seems like you are running down a rabbit hole.

As before, you come back to this notion of tying an expression that can only be solved numerically to whether or not that expression can be considered to be in closed form.

So, as before, I'll ask again.

If we have

y = cos(x)

and I tell you that x is 2.57, how are you going to evaluate that expression to find y other than numerically? Except for special case values of x, we can only evaluate cos(x) to an approximate result numerically using a power series. The same for ln(x), exp(x) and many, many others. So if it can only be solved numerically and that is your criteria for being in closed form, then you can't claim that this equation is in closed form.

But the pretty much universally accepted definition of what a closed form solution is is the one given by Wolfram, which would pretty clearly say that this IS a closed form solution.

Think of it another way.

Let's say that I make a new calculator with signal processing in mind. Just as I have sin() and arcsin() functions on their, I would probably have sinc() and arcsinc() functions on there.

Now let's say that I want to find the value of y such that

y = x·sin(y)

for a given value of x.

You are saying that this is of the form y = f(x,y).

But I can write it as

x = y/sin(y) = sinc(y)

and solve this as

y = arcsinc(x)

Now I plug the value of x into my calculator and hit the arcsinc() button and I have my answer.

How is this ANY different than solving y = cos(x)?

 

You suggested the following:

y = x·cos(y)

I solved it for y symbolically by defining a function fred() and it's inverse.

There is fundamentally NO difference between using fred() to solve it symbolically than using cos(), ln(), exp(), or any other power series function and/or their inverses.



How is reducing the infinite sum for cos(x) to cos(x) any different than reducing the infinite sum for fred(x) to fred(x) or the infinite sum for sinc(x) to sinc(x) or the infinite sum for ln(x) to ln(x)?.

Fundamentally, the only difference is that cos(x) is a name for a power series function that essentially everyone agrees is a "well-known" or "generally-accepted" function for the purposes of deciding which functions can appear in closed-form solutions, which fred(x) likely isn't. But, as Wolfram states explicitly, this is an inherently arbitrary decision.



Which is the major point of this side discussion -- which functions and which operations are considered "generally-accepted" or "well-known"? Who decides? The Wolfram entry explicitly points out that there is no universal agreement and that different branches of mathematics have different sets.

Maybe a limited power series is a closed form but an infinite series cant be. This seems like a good way to look at it because:
y=1+x+x^2+x^3
is closed but then
y=1+x+x^2+x^3+...+x^n
is hard to call closed.

Let me reiterate the problem i've seen again.
We have a function:
y=f(x)

and it is easy to use in another function:
z=g(f(x))

but if we have:
y=f(x,y)

we cant put it into that same function g() because we would end up with:
z=g(f(x,y))
which can never be solved because we dont know what y is, unless as before we can actually solve for it, but when we cant not that is when the problem comes up certainly not when we can.

Take this function:
x^y=e^y+e^(-y)

We can solve this for x but can we solve this for y? Maybe, but let's say we cant. Then how do we calculate:
z=2*y+3

First, we are implying that we end up with:
z=2*f(x)+3

but we dont know that the function f(x) is. To solve, we have to know what x is first, then solve the first equation for y, then insert that numerical value into the expression for z.

Note that the function you substituted for mine is able to be solved so that's not the same thing.

This kind of problem pops up a lot. One i remember best is the exact solution to the full wave (or half wave) rectifier with filter cap and resistive load with maybe a series resistance too and solving based on the voltage responses. The expression that comes up can only be solved numerically. Can it still be said to be in closed form?

Maybe it would be better to just abandon that classification because it is sometimes too subject to interpretation.

It seems like you are running down a rabbit hole.

As before, you come back to this notion of tying an expression that can only be solved numerically to whether or not that expression can be considered to be in closed form.

So, as before, I'll ask again.

If we have

y = cos(x)

and I tell you that x is 2.57, how are you going to evaluate that expression to find y other than numerically? Except for special case values of x, we can only evaluate cos(x) to an approximate result numerically using a power series. The same for ln(x), exp(x) and many, many others. So if it can only be solved numerically and that is your criteria for being in closed form, then you can't claim that this equation is in closed form.

But the pretty much universally accepted definition of what a closed form solution is is the one given by Wolfram, which would pretty clearly say that this IS a closed form solution.

Think of it another way.

Let's say that I make a new calculator with signal processing in mind. Just as I have sin() and arcsin() functions on their, I would probably have sinc() and arcsinc() functions on there.

Now let's say that I want to find the value of y such that

y = x·sin(y)

for a given value of x.

You are saying that this is of the form y = f(x,y).

But I can write it as

x = y/sin(y) = sinc(y)

and solve this as

y = arcsinc(x)

Now I plug the value of x into my calculator and hit the arcsinc() button and I have my answer.

How is this ANY different than solving y = cos(x)?[/QUOTE]


Hi,

To be honest, im not sure what you are saying here and i dont think you appreciate my explanation because you seem to be suggesting that everything can be solved symbolically while it's not so. Do i misunderstand something?

The function:
y=cos(x)

and we want to put that into
z=g(y)

we just take cos(x) and put in in place of y and get:
z=g(cos(x))

That i would consider closed.

But there are some equations that can only be written in implicit form:
y=f(x,y)

and with this class are not be able to solve for y symbolically. We must know the value of x numerically and then solve for y numerically, and then we can put that into the function:
z=g(y)

So say we used x1 and got the value y1, then we would end up with:
z=g(y1)

that's obvious. But because the first equation can only be written in implicit form, we cant come out with a symbolic solution:
z=g(something)

like say:
z=something ^3

because we could never compute "something" symbolically.
The only solutions we would be able to come up with would be like:
z=1.2^3 or z=4.567^3, etc.

there's never going to be a solution like:
z=(x^2+2*x+1.258)^3

because x has to be specified numerically to solve the first equation.

So the first equation i would not called closed.

Yes sure you can solve some equations symbolically, but you can not do some that way. Let me see if i can find one that would show better.

 

Hello again,

Ok here is one from another thread that looks hard to solve:

(51.264*v1^0.5054+32.04*(120-v1)^0.5054)*v1-6151.68*v1^0.5054=0

which we can write as:
f(v1)=0

we need to solve that for v1. We can then use it in:
y=g(v1)

where g may be complicated but symbolically solvable such as:
g(x)=x/R1

or not:
g(x)=x/R1(x)

or also:
g(x)=120/(R1(x)+R2(x))

where these two resistances depend on x which equals v1.

I dont think any of these can be said to be in closed form but take a look.
My main point being that although we may not be able to specify what is in closed form, we probably can often specify that something specific is NOT in closed form. To put it another way, sometimes we have a result that is too arbitrary and complicated enough to say it is not in closed form. At a later date, someone my stumble on a solution that is in closed form, but until then it's not.

 

On page 4 and 5 of the Borwein and Crandall paper, the LambertW is described as "well studied", and a "...splendid closed form..."

https://www.carma.newcastle.edu.au/jon/closed-form.pdf

 

On page 4 and 5 of the Borwein and Crandall paper, the LambertW is described as "well studied", and a "...splendid closed form..."

https://www.carma.newcastle.edu.au/jon/closed-form.pdf

View attachment 172962

Hi again,

Thanks for the paper that helps understand this interesting question better.

Because of the debacle over what was considered a closed form i decided to just address the question of what was NOT considered a closed form. Sounds the same but it's not because i think we can show some examples of what can not be considered closed form.
That's still not the end of the story though. Even though we could show examples of what can not be considered to be in closed form. it may be possible that a change in variable or topical space transformation can turn it into a closed form. However, that would end up as a different equation or expression so i dont think we can say that the original is in closed form even though the transformed one is.

 

Maybe it is always subject to interpretation? Just like what we call 'linear' is subject to a judgement based on the independent variable.

What? Depending on context, linear has two well-defined meanings, neither of which are subjective. In one context, we call first-degree polynomials linear; in another, a function f is called linear if and only if it satisfies two conditions: f(cx) = c f(x), for some constant c, and f(x+y) = f(x) + f(y).

The epithet closed-form, on the other hand, is completely subjective. Is the gamma function closed-form? Depends on whom you ask.

Maybe a limited power series is a closed form but an infinite series cant be.

Then you can't say that f(x) = e^x is closed-form, which then precludes ln(x) and all of the trig functions.

 

In my reply was:
"Maybe it is always subject to interpretation? Just like what we call 'linear' is subject to a judgement based on the independent variable."

What? Depending on context, linear has two well-defined meanings, neither of which are subjective. In one context, we call first-degree polynomials linear; in another, a function f is called linear if and only if it satisfies two conditions: f(cx) = c f(x), for some constant c, and f(x+y) = f(x) + f(y).

Yeah it looks like you dont understand how overall context can change. You're looking at the detail of the definition of linear and say that linear can not change. But that's an internal view only. You seem to approach questionable things this way. It's like talking about a tree and you can see the limbs and leaves, but you cant see how it fits in with the rest of the trees. In fact, it's like you have no idea that there are even any other trees around.
Functions can be linear in one domain and not linear in some other domain and we see this all the time.

The epithet closed-form, on the other hand, is completely subjective. Is the gamma function closed-form? Depends on whom you ask.
Then you can't say that f(x) = e^x is closed-form, which then precludes ln(x) and all of the trig functions.

Yeah but i was already hinting at that as being one of the conclusions. I since then moved on. I moved on to trying to define what is NOT in closed form. I think we can define things as being not in closed form with the restriction that we know there could be a future solution that we just dont know of yet.

So anyway, think about those two ideas, and thank you for the reply.

 

Interesting read out about Lambert W function and semiconductors
https://paklaunchsite.jimdo.com/app/download/7697335554/PAK-Course-101-117-ALL.pdf?t=1553235575
https://paklaunchsite.jimdo.com/pak2/

 

Yeah it looks like you dont understand how overall context can change. You're looking at the detail of the definition of linear and say that linear can not change. But that's an internal view only. You seem to approach questionable things this way. It's like talking about a tree and you can see the limbs and leaves, but you cant see how it fits in with the rest of the trees. In fact, it's like you have no idea that there are even any other trees around.

You were making a point about the definition of closed-form, suggesting that it may be subjective as -- according to you -- is the case for linearity. I replied that linearity is precisely defined; what context am I missing?

Functions can be linear in one domain and not linear in some other domain and we see this all the time.

This is nonsensical to me. A function is either linear or it is not; if we change the domain, then it is no longer the same function, so we wouldn't expect its linearity to somehow carry over. If you insist that linearity is subjective, then please give an example that two mathematicians might rightly disagree on.

I moved on to trying to define what is NOT in closed form.

I don't see how that helps, as whatever set you end up with, just take its complement.

What's wrong with simply saying that closed-form expressions are those that are either convenient to write down (e.g., polynomials) or have been made convenient to write down because they are commonly used (e.g., trig functions)? What exactly is considered common will vary amongst different mathematical fields, so it is indeed subjective, but who cares?

 

You were making a point about the definition of closed-form, suggesting that it may be subjective as -- according to you -- is the case for linearity. I replied that linearity is precisely defined; what context am I missing?


This is nonsensical to me. A function is either linear or it is not; if we change the domain, then it is no longer the same function, so we wouldn't expect its linearity to somehow carry over. If you insist that linearity is subjective, then please give an example that two mathematicians might rightly disagree on.


I don't see how that helps, as whatever set you end up with, just take its complement.

What's wrong with simply saying that closed-form expressions are those that are either convenient to write down (e.g., polynomials) or have been made convenient to write down because they are commonly used (e.g., trig functions)? What exactly is considered common will vary amongst different mathematical fields, so it is indeed subjective, but who cares?

Hello again,

You mean you have never heard or read of anyone that said that sin(wt) is linear? Even though in the time domain it is not?