Math Is Fun Forum

GOKILL

Member

Registered: 2010-03-19

Posts: 26

Unusual equality :))

1. Find the solution of

(i think there's no solution, but i'm not sure how to prove it analitically )

2. Find the area between the curves

and

(i feel the difficulty was in finding the point where they're crossing over )

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mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Unusual equality :))

Not sure if this is rigourous enough for you, but:
e^x > 0 for all x, so clearly there is no negative solution.
d(e^x)/dx = e^x, which is >1 for all x>0. Since e^0 = 1 > 0 and dx/dx = 1, there can't be any solution greater than 0 either. Hence there is no solution to e^x = x.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Unusual equality :))

mathsyperson, I believe #1 is supposed to be a question in complex analysis (this is why z is used).

As to the answer, it's an application of Rouche's theorem.  The "hint" below is really an answer...

This proves that there is exactly one value where e^z = z.

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GOKILL

Member

Registered: 2010-03-19

Posts: 26

Re: Unusual equality :))

at first i think it was in real, so there's no solution.
i don't know how it would be in case of z is in complex..
so would u explain f(z) = e^z - z, g(z) = z ?
what's the answer?

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juriguen

Member

Registered: 2009-07-05

Posts: 59

Re: Unusual equality :))

Hi there!

I actually got help from quickmath.com, and then checked what did the result mean. The idea is to use what is called the Lambert W function, which, by definition (Wikipedia) allows to express any complex number as:

Then, for your equation you have to do:

But any number can be written using the W function, so:

By equating both expressions, then:

which actually has |z| = 1.3745, so the condition is satisfied.

The other problem I cannot get it yet. I will keep trying

Ok, so I would say you have to do this numerically. First plot both functions, to have an idea. then use Newton-Rapson's method to obtain when they intersect. For instance, to calculate the area in the interval [0, 3pi/2], first lets find where they cross:

Say we want to find

Using N-R, from a guess xn, we obtain a better guess doing:

Choose as x0 = 4.5, and a few iterations give x = 4.4934

I have calculated only the area between [0, 4.4934], which is:

Hope this helps!
Jose

Last edited by juriguen (2010-03-20 11:07:39)

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“Make everything as simple as possible, but not simpler.” -- Albert Einstein

scientia

Member

Registered: 2009-11-13

Posts: 224

Re: Unusual equality :))

I actually got help from quickmath.com, and then checked what did the result mean. The idea is to use what is called the Lambert W function, which, by definition (Wikipedia) allows to express any complex number as:

Then, for your equation you have to do:

But any number can be written using the W function, so:

By equating both expressions, then:

which actually has |z| = 1.3745, so the condition is satisfied.

This only proves that a solution exists. I think the OP wants to find all complex solutions of the equation.

juriguen

Member

Registered: 2009-07-05

Posts: 59

Re: Unusual equality :))

Hi!

I think there's only 1 solution, and I wrote it down, z = -W(-1)

Right?

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“Make everything as simple as possible, but not simpler.” -- Albert Einstein

GOKILL

Member

Registered: 2010-03-19

Posts: 26

Re: Unusual equality :))

Some one has sugest me to use rouche's theorem but i don't know bout it >_<

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I am the greatest magician this century!!!