Does this make the function Transcendental?

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In summary, the conversation is about Euler's Introduction to Analysis of the Infinite and the distinction between algebraic and transcendental functions. The question is raised whether the inability to write a function explicitly is a cause of "transcendentality". The conversation is then moved to the topic of calculus and analysis. The answer is provided through a reference to the definition of transcendental functions, which states that a function is transcendental if it cannot be expressed by a finite sequence of algebraic operations.
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sponsoredwalk
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Hi I'm reading Euler's Introduction to Analysis of the Infinite & he's distinguishing between algebraic & transcendental functions and I wonder is the fact that the following function cannot be written explicitly a cause of "transcendentality".

[tex] Z^5 = az^2Z^3 - bz^4Z^2 + cz^3Z - 1[/tex]

where Z is a function of z.

Maybe I've read it wrong or asked the question too soon but does being unable to write a function explicitly automatically make the answer transcendental?
 
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I have moved this thread to "Calculus and Analysis" because it is question, not "learning materials".
 
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1. Does having a non-algebraic expression in the function make it transcendental?

No, a function can still be algebraic even if it contains non-algebraic expressions, as long as it can be expressed using a finite number of algebraic operations.

2. Can a function be both algebraic and transcendental?

No, a function can only be either algebraic or transcendental, not both. A function is considered transcendental if it cannot be expressed using a finite number of algebraic operations.

3. Does the presence of transcendental constants in the function make it transcendental?

Yes, if a function contains transcendental constants (such as pi or e) and cannot be expressed using a finite number of algebraic operations, it is considered transcendental.

4. How can I determine if a function is transcendental?

A function is transcendental if it cannot be expressed using a finite number of algebraic operations. In other words, if the function involves non-algebraic operations (such as logarithms, exponentials, and trigonometric functions) or transcendental constants, it is considered transcendental.

5. Is there a specific set of rules or criteria for determining if a function is transcendental?

Yes, there are certain rules and criteria that can be used to determine if a function is transcendental. These include checking if the function involves non-algebraic operations or transcendental constants, and if it satisfies certain properties such as being continuous and having infinitely many solutions. However, it ultimately depends on the specific function and there is no one-size-fits-all approach.

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