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I looked up several sources on the internet.

A transcendental equation is an equation containing a transcendental function of the variable(s) being solved for. Such equations often do not have closed-form solutions.

And then transcendental function

A transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.

Unfortunately, I don't quite understand what it means "to satisfy polynomial equation"

I also checked several questions asked on this site. One of the most relevant is:

In simple English, what does it mean to be transcendental?

However, although OP asked about "transcendental function in layman terms" , the most pertinent answers mostly answer the question "What is transcendental number"

So I would like to ask you, if we use the most basic language possible, what is transcendental equation/function? And how do I determine whether one is a transcendental function/equation?

Stokolos Ilya
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  • If you read the tag descriptions then you would have seen that "Transcendental equations are equations containing transcendental functions, i.e. functions which are not algebraic. An algebraic function is a function that satisfies a polynomial equation whose terms are themselves polynomials with rational coefficients." – Peter Foreman Aug 25 '19 at 16:26
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    For instance, $f(x)=\sqrt x$ is algebraic because it satisfies $\left( f(x)\right)^2-x=0$. And $g(x)=\frac {x-1}{x^2}$ is algebraic because it satisfies $x^2g(x)-(x-1)=0$. – lulu Aug 25 '19 at 16:27
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    Notice that the function $f(x)= \sqrt{x^2- 2x+ 1}$ satisfies $f^2= x^2- 2x+ 1$. That function "satisfies a polynomial equation". More simply, a function is "transcendental" if it cannot be written as combinations (sums, products, quotients, compositions) of polynomials and roots. – user247327 Aug 25 '19 at 16:29
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    Honestly, I am struggling to see how this question is not answered by the answers to the linked question. Yes, those answers talk about transcendental numbers more than functions, but I think that the question has been answered. – Xander Henderson Aug 25 '19 at 16:54
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    What about the second paragraph of this answer https://math.stackexchange.com/a/1686170 ? – quid Aug 25 '19 at 20:53

2 Answers2

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A polynomial is an expression obtained by combining constants and variables by means of a finite number of additions and multiplications.

E.g. $3xy^3+2x-1$.

An algebraic function of a single variable $x$ is such that the relation to the dependent variable $y(x)$ can be expressed by a bivariate polynomial equation with integer coefficients.

E.g. $3x(y(x))^3+2x-1=0$, which can also be written $y(x)=\sqrt[3]{\dfrac{1-2x}{3x}}$.

In particular, the ratio of two polynomials in $x$ is an algebraic function, as is any expression involving only the four operations and radicals.

A transcendental function is one that is not algebraic.

E.g. $\sin(x)$ is transcendental because there is no polynomial $P$ such that $P(x,\sin(x))=0$.

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a transcendental function f(x) gives transcendental results for most rational x example: e^x, sin(x) etc. the simple seaming equation e^x=x or cos(x)=x have no formula for x as result, but must be calculated numerically. also you cn not rewrite e^x as a polynomial or a fraction of polynoms trula

trula
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    A priori, the transcendentalness of a function is unrelated to the transcendentalness of its outputs. – Arthur Aug 25 '19 at 16:39
  • @Arthur: not really. An algebraic function takes algebraic values when the argument is algebraic. Only transcendental functions can "create" transcendental numbers. –  Jun 08 '21 at 13:57