Intervals with Transcendental Outputs

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In summary, the conversation discusses the possibility of testing when a function will have transcendental outputs versus rational or algebraic outputs. Specifically, the function ssrt(x) is mentioned, which is the inverse of a second degree power tower. The conversation then mentions the use of Lambert's W-function and the Gelfond-Schneider theorem to find x intervals that make the function f(x) transcendental.
  • #1
Hertz
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Maybe I'm getting a bit ahead of myself here, but for the sake of curiosity I'll ask it anyways. Is there any way that you can test when a function h(x) will have transcendental outputs vs rational or algebraic outputs?

Specifically, if I have the following function
f(x) = ssrt(x)

Where ssrt is the inverse of a second degree power tower (ssrt(x^x) = x).
Meaning, for a given value of x, the ssrt() function will give me the number that when raised to the power of itself equals x.

Good examples of this function are ssrt(4) = 2 because 2^2 = 4 and ssrt(27) = 3 because 3^3 = 27.​

Would it be possible to find the x intervals that make f(x) transcendental?
 
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  • #2
I doubt that there are intervals as solution. Generally you should look up Lambert's W-function and the Gelfond-Schneider theorem.
 

Related to Intervals with Transcendental Outputs

1. What are intervals with transcendental outputs?

Intervals with transcendental outputs refer to a set of values or numbers within a specific range that cannot be expressed as a ratio of two integers. These values are known as transcendental numbers and include pi, e, and square roots of prime numbers. Intervals with transcendental outputs are used in various mathematical and scientific calculations.

2. How are intervals with transcendental outputs different from intervals with rational outputs?

The main difference between intervals with transcendental outputs and intervals with rational outputs is that transcendental numbers cannot be expressed as a ratio of two integers, while rational numbers can. This means that intervals with transcendental outputs will have decimal expansions that do not terminate or repeat, while intervals with rational outputs will have decimal expansions that eventually end or repeat.

3. What are some applications of intervals with transcendental outputs?

Intervals with transcendental outputs are used in various mathematical and scientific fields, such as calculus, physics, and engineering. They are also used in computer science for tasks such as generating random numbers and cryptography. Additionally, transcendental numbers have been used in art and music to create aesthetically pleasing patterns and melodies.

4. How are intervals with transcendental outputs calculated and represented?

Intervals with transcendental outputs are usually calculated using mathematical formulas or algorithms, depending on the specific application. They can be represented in different ways, such as using decimal approximations, continued fractions, or symbolic notations. In some cases, intervals with transcendental outputs may also be graphed on a number line or displayed in a table.

5. Are there any real-life examples of intervals with transcendental outputs?

Yes, there are many real-life examples of intervals with transcendental outputs. For instance, the circumference of a circle, which is represented by the transcendental number pi, is an interval with a transcendental output. Other examples include the decay rate of radioactive substances, the value of the natural logarithm, and the wavelength of light in a vacuum. These intervals with transcendental outputs are essential in understanding and predicting natural phenomena.

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