What Happens When You Differentiate Euler's Formula?

In summary: So my understanding of decimation in time is to perhaps remove every other sample or every third sample or every fourth sample...The Fast Fourier Transform (FFT) algorithm is a way to efficiently compute the discrete Fourier transform (DFT) of a sequence of numbers. It's called "fast" because it can compute the DFT much faster than using the definition of the DFT, which involves a lot of complex multiplication.The FFT algorithm is based on a divide-and-conquer approach, where the sequence of numbers is recursively split into smaller and smaller sub-sequences until each sub-sequence contains only one number. Then, the DFT of each sub-sequence is computed and combined to
  • #1
ADDA
67
2
https://en.wikipedia.org/wiki/Euler's_formula

(1) eix = cos(x) + isin(x)
(2) eixidx = (-sin(x) + icos(x))dx
(3) eix = (-sin(x) + icos(x)) / i
(4) eix = cos(x) + isin(x)
Just lost in circles.

Just for fun.. post a solution for x.
 
Last edited:
Physics news on Phys.org
  • #2
ADDA said:
https://en.wikipedia.org/wiki/Euler's_formula

(1) eix = cos(x) + isin(x)
(2) eixidx = (-sin(x) + icos(x))dx
(3) eix = (-sin(x) + icos(x)) / i
(4) eix = cos(x) + isin(x)
Just lost in circles.
Technically, you didn't differentiate equation 1 -- you took the differential of each side.
If you differentate equation 1 with respect to x, you get
(2) eixi = -sin(x) + icos(x)

If you divide both sides by i, and do the usual trick of multiplying ##\frac{-\sin(x)}{i}## by ##\frac {-i}{-i}##, you get what you would expect.
ADDA said:
Just for fun.. post a solution for x.
Euler's formula is an identity -- it's true for all x, so it's not possible to solve for x.

Added: We know that ##\frac d {dx} e^x = e^x##, and you have found that ##\frac d {dx} e^{ix} = e^{ix}## as well. Is that so surprising?
 
Last edited:
  • #3
Is there a question?
 
  • #4
Take item (3) and square both sides:

e2ix = -1 * (-sin(x) + icos(x))2

expanding the right side:

sin2(x) - i2sin(x)cos(x) - cos2(x)

apply the double angle identity

- isin(2x) - (cos2(x) - sin2(x))

apply the double angle identity

-cos(2x) - isin(2x)

Did I miss the sign above?
 
  • #5
ADDA said:
Take item (3) and square both sides:

e2ix = -1 * (-sin(x) + icos(x))2

expanding the right side:

sin2(x) - i2sin(x)cos(x) - cos2(x)
No.
You have two sign errors in the line above.

You should have gotten ##-\sin^2(x) + 2i \sin(x)\cos(x) + \cos^2(x)##, which can be simplified to ##\cos^2(x) - \sin^2(x) + i \cdot 2\sin(x)\cos(x) = \cos(2x) + i \sin(2x)##.
From Euler's Formula directly, ##e^{2ix} = \cos(2x) + i\sin(2x)##
ADDA said:
apply the double angle identity

- isin(2x) - (cos2(x) - sin2(x))

apply the double angle identity

-cos(2x) - isin(2x)

Did I miss the sign above?
 
  • #6
Thanks Mark44, I've been experimenting with trying to change the radix of a fast Fourier transform. My solution for x is this:

Since squaring both sides could equal:

eix2 = eix2

equating the exponents of e would yield

ix2 = i * x * 2;

-x2 = 2 * i * x;

so x = -2 * i.

Now if I wanted to do a radix three fast Fourier transform, would I cube both sides of item three above?

eix3 = ((-sin(x) + icos(x)) / i)3
 
  • #7
ADDA said:
Thanks Mark44, I've been experimenting with trying to change the radix of a fast Fourier transform. My solution for x is this:

Since squaring both sides could equal:

eix2 = eix2
No. Squaring ##e^{ix}## does not result in ##e^{ix^2}##.
Squaring ##e^{ix}## results in ##\left(e^{ix}\right)^2## which can be simplified to ##e^{2ix}##
ADDA said:
equating the exponents of e would yield

ix2 = i * x * 2;
No, since you have a mistake in your previous work. Taking out the factor of i, you're saying that ##x^2 = 2x##, which is true only for x = 2 or x = 0.
ADDA said:
-x2 = 2 * i * x;

so x = -2 * i.
Again, no, for the reasons stated above.
ADDA said:
Now if I wanted to do a radix three fast Fourier transform, would I cube both sides of item three above?

eix3 = ((-sin(x) + icos(x)) / i)3
Why don't you back up and see if you can get the work right for squaring? You've had several mistakes throughout this thread.
 
Last edited:
  • Like
Likes Gregory Alan
  • #8
"To the extent that a fool knows his foolishness, He may be deemed wise. A fool who considers himself wise is indeed a fool."

Thanks again Mark44, modens ponens.

One other question, say I perform a radix 9 transform, in the transform data, would a peak magnitude index represent 1.0 / 9.0 of a usual radix two index which would be 1.0 / 2.0 the transform data?

My discrete formula is this:

index = (frequency * frame_size) / sample_rate

so I'm thinking I would need

index = (frequency * frame_size) / (radix * sample_rate)

I'm able to somewhat reproduce the original audio, and have a video example. Is there a way to upload a video to your site without an external link?
 
Last edited:
  • #9
ADDA said:
One other question, say I perform a radix 9 transform, in the transform data, would a peak magnitude index represent 1.0 / 9.0 of a usual radix two index which would be 1.0 / 2.0 the transform data?

My discrete formula is this:

index = (frequency * frame_size) / sample_rate

so I'm thinking I would need

index = (frequency * frame_size) / (radix * sample_rate)
I'm not seeing what this has to do with your original question about Euler's formula. The word "radix" has different meanings, with one of them being the symbol √ for square roots, but more commonly referred to as "radical sign."

Are you asking about taking square roots, cube roots, ninth roots, etc.?
 
  • #10
Technically, again I am wrong. I looked back in the book. A decimation in time fast Fourier transform starts with many radix-2 transforms then many radix-4 transforms, then many radix-8, continually until frame size / 2 is reached. So I'm attempting a base 3 or base 9 or base 5 transform, I've just termed it radix. So, I've modified the process to allow many radix-3 transforms to many radix-9 transforms to many radix-27 transforms...
 
  • #11
ADDA said:
Technically, again I am wrong. I looked back in the book. A decimation in time fast Fourier transform starts with many radix-2 transforms then many radix-4 transforms, then many radix-8, continually until frame size / 2 is reached. So I'm attempting a base 3 or base 9 or base 5 transform, I've just termed it radix. So, I've modified the process to allow many radix-3 transforms to many radix-9 transforms to many radix-27 transforms...
You haven't explained what you mean by a radix-2/radix-4-radix-8 transform, nor what you mean by a base-3, base-9, or base-5 transform, so I have no idea what you are saying here.

Also what do you mean by "decimation in time fast Fourier transform"?
The literal meaning of decimation is the removal of every 10th element.
 
  • #12
ADDA said:
Technically, again I am wrong. I looked back in the book. A decimation in time fast Fourier transform starts with many radix-2 transforms then many radix-4 transforms, then many radix-8, continually until frame size / 2 is reached. So I'm attempting a base 3 or base 9 or base 5 transform, I've just termed it radix. So, I've modified the process to allow many radix-3 transforms to many radix-9 transforms to many radix-27 transforms...
Are you talking somehow about a discrete FFT?
 

Related to What Happens When You Differentiate Euler's Formula?

What is Euler's formula?

Euler's formula is a mathematical formula that relates the exponential function, complex numbers, and trigonometric functions.

What is the derivative of Euler's formula?

The derivative of Euler's formula is eix where i is the imaginary unit. This means that the derivative of Euler's formula is itself.

How is Euler's formula used in science?

Euler's formula is used in various fields of science, such as physics, engineering, and mathematics. It is used to simplify complex calculations involving exponential and trigonometric functions.

What is the connection between Euler's formula and the Unit Circle?

Euler's formula can be used to represent complex numbers on the unit circle, which is a circle with a radius of 1 centered at the origin on a Cartesian plane. This allows for a geometric interpretation of complex numbers and their relationships with trigonometric functions.

What is the significance of Euler's formula in mathematics?

Euler's formula is significant in mathematics because it provides a fundamental connection between exponential and trigonometric functions. It also has applications in various fields of mathematics such as complex analysis, number theory, and differential equations.

Similar threads

Replies
4
Views
879
Replies
4
Views
2K
  • Calculus
Replies
4
Views
2K
  • Differential Equations
Replies
3
Views
2K
Replies
2
Views
1K
  • Calculus
Replies
7
Views
2K
Replies
5
Views
1K
Replies
1
Views
1K
  • Differential Equations
Replies
2
Views
1K
Back
Top