This is a terrible article. I reads like an answer to "Explain a transcendental equation off the top of your head, right now!" -98.228.254.203 (talk) 05:13, 9 May 2011 (UTC)[reply]
I totally agree. But currently we have few ways to do with transcendent cases. --IkamusumeFan (talk) 05:59, 3 November 2014 (UTC)[reply]
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Use a multivariate function to express the solution of a general transcendental equation
Please read 'Use a multivariate function to express the solution of a general transcendental equation', Is it easy to be understand? Can you accept it?You are welcome to improve it.
Note: Multivariate function composition and inverse multivariate function see below.
The expression to the solution of the equation is:
It is enough for you to know how to obtain the expression of the solution for a given
binary operators
being composition and
unary operators
such as promotion or oblique projection or inverses .
Solving an equation is reducing several X to one then putting the X on one side of '=' and putting all the known things on the other side. The expression shown here meets this requirement.Is the expression a real solution? We obtain this expression by three steps,function promotion,multivariate function composition and multivariate inverse function. Which step can not be accepted by us? Function promotion? It is just changing binary operations or binary functions as special ones of n variables. Multivariatefunction composition? There is unary function composition. Why is no there multivariate function composition? In the same reason, there is unary inverse function, there must be multivariate inverse function! So I can not find any reason to reject such an expression.
Why do we use these forms? We can describe any expression in a fire-new way. For example,,first we denote it as , in which and . In addition, we denote subtraction as ,multiplication as , division as , root as and logarithm as respectively. We want give an expression like in which the left part is called bare function containing only symbolics of function and the right part contains only variables.
is an expression of a function of three variables. We consider and as especial
By these examples we know the meaning of superscript and subscript of and we call it function promotion.
It is clear that we obtain by substituting and in by and respectively. So can be written in:
or
or
We never mind how complex they are. We consider them as
multivariate functions
being composition results and or promotion results. These new expressions are different from . Actually we had departed bare function from variables in these new expressions and there is only one "x" in them. This is what we want to do when we solve transcendental equations like .
For an unary function promotion, . In special,, in which 'e' is the identity function.
In if and
Note,there is no in the expression.
is called oblique projection of f. Actually it is a function of n-1 variables and is dependent on only f and i,j so we denote it as .
For example,
Inverse multivariate function
For multivariate function ,.
If is bijection for any we call an multivariate inverse function about . Introduce unary operator and denote :
.
For example, is invertible about variable and is not invertible about variable .
Partial inverses can be extend to
multivariate functions
too. We can define multivariate inverse function for an irreversible function if we can divide it into r partial functions and denote its inverses as :
@Jochen Burghardt: Defining a transcendental equation as one where one or both sides are transcendental functions is problematic. That would mean that and are transcendental equations (both sides are transcendental functions), but the equivalent and are not (neither side is). Determining whether an expression represents a transcendental function is non-trivial in general. Is only a transcendental equation for some values of ?
Other sources agree that a transcendental equation is one which involves transcendental functions.[1] --Macrakis (talk) 20:29, 6 January 2022 (UTC)[reply]
@Macrakis and D.Lazard: I just checked Bronstein et al. again; they say "Eine Gleichung F(x) = f(x) ist transzendent, wenn wenigstens eine der Funktionen F(x) oder f(x) nicht algebraisch ist." "An equation F(x) = f(x) is transcendental if at least one of the functions F(x) or f(x) is not algebraic." So there are different definitions around (I'm not sure that Bronstein et al. are aware of the difference).
Maybe "involves" is the best choice for the definition, per your above arguments. The equation would then be considered a transcendental one, which could trivially be transformed into an algebraic one. This is similar to which is not a quadratic equation in the strict sense, but can be trivially transformed into one. There are many different degrees of transformation difficulty, while a definition should provide a sharp yes/no criterion. So, if there are no objections, I'd change the definition to that of Macrakis/mathworld, and add a remark about Bronstein's deviation. - Jochen Burghardt (talk) 21:39, 6 January 2022 (UTC)[reply]