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[WP34s] Regularized incomplete Beta function
05-03-2014, 01:38 PM
Post: #19
RE: [WP34s] Regularized incomplete Beta function
This is actually funny. You usually call the first complex variable z, and the first real variable x. As the definitions of the gamma/beta functions start with real functions and then get analytically continued to the complex plane, it is implied that a definition with x is for real arguments and if we considered complex arguments, it would involve z instead.

The Abramowitz notation is for real values of x, because only real values are considered there. The current DLMF from NIST uses x because it is assumed that a>0, b>0, 0<=x<=1, and then "it is straightforward to continue most results analytically to other real values of a, b, and x, and also to complex values."

The Wolfram notation is just this, with z, as the Mathematica implementation uses complex values.

But alas, the stack has levels traditionally labelled x, y, z, t. Well, they're just levels 1, 2, 3, 4 with input sequence {4, 3, 2, 1}. Label names don't have to mean anything. Whatever complex or real variable is considered: x or z, the variable name by itself shouldn't determine the position in the stack.


If you enter things as they appear in standard notation you have the y^x ordering, that is var1^var2 has the input sequence {var1, var2} and a stack [x = var2, y = var1, z = 0, t = 0].

This way I("x or z"; a, b) would be I(var1; var2, var3) with input sequence {var1, var2, var3} and a stack [x = var3, y = var2, z = var1, t = 0].

Now, if we want to give the first position in the stack to the "main" variable, that is, if we consider the other variables as parameters, then I("x or z"; a, b) would have two choices for the stack [x = "x or z", y = a, z = b, t = 0] or [x = "x or z", y = b, z = a, t = 0]. There's no way to choose one of them as more logical than the other unless we adopt another convention. And that would likely be that the values in the stack must have the same order in which they appear in the standard notation left to right written expression, i.e. the first one. The input sequence would be: {b, a, "x or z"}.

According to this, if we consider the function with value var1^var2 and we think that the "main" variable is var1, our printed y^x is wrong and should be x^y. If the "main" variable is var2 it's all fine.

As this is a source of potentially unending confusion, at least for variable lists I'd just enter things as they appear in written expressions, independently of the names of the variables.
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RE: [WP34s] Regularized incomplete Beta function - Manolo Sobrino - 05-03-2014 01:38 PM



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