Kitonum

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15 years, 237 days

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These are questions asked by Kitonum

How to simplify this trigonometric expression with Maple? I only know the way with  identify command, which is difficult to call mathematically correct:

restart;
Expr:=arctan((1-tan(20*Pi/180))/(1-tan(25*Pi/180)));
evalf[15](Expr);
identify(%);

                           

        

 

The problem: to simplify the expression

for any negative  x  and  y .

Below we see that Maple copes with the task brilliantly (example 1). For example, it presents  sqrt(x*y)  as  sqrt(-x)*sqrt(-y)  and so on. But the same technique, applied only to the numerator of this expression does not give the desired presentation in the form of a square (example 2 and example 3).

restart;
# Example 1
A:=(x+y-2*sqrt(x*y))/(sqrt(-x)+sqrt(-y));
simplify(A) assuming negative;
factor(%,{sqrt(-x),sqrt(-y)});

                                         

restart;
# Example 2
B:=x+y-2*sqrt(x*y);
simplify(B) assuming negative;
factor(%,{sqrt(-x),sqrt(-y)});

                                           

 

restart;
# Example 3
B:=x+y-2*sqrt(x*y);
R:=simplify(B) assuming positive;
combine(R) assuming positive;
factor(R,{sqrt(x),sqrt(y)});

                                                   

Two questions:

1. Does anyone know the reasons for this behavior.

2. Does anyone know an easy way to simplify in examples 2 and 3 (without  substitutions  like  x=+-u^2  and  y=+-v^2 and so on,  of course)

 

Download sqrt.mw

 

after running the command  DocumentTools:-Tabulate  as you can see it in the example below in Maple 2018.2:

DocumentTools:-Tabulate(Matrix(2, [a, b, c, d]), width=20);

Output:
                                  

 

 

An hour or two ago, I answered a question in which it was a question of plotting a complex-valued function of 2 real variables. But the question itself and also my answer to it disappeared somewhere. Therefore, I send my answer here below.

There are two options for plotting:
1. Graphs of real and imaginary parts (as 2 surfaces in 3D).
2. Graph of the absolute value of this function (one surface in 3d) .

restart;
f:=(1+cosh(2*x))*exp(-4*I*t):
plot3d([Re,Im](f), x=0..1, t=0..1, color=[red,blue]);
plot3d(sqrt(add([Re,Im](f)^~2)), x=0..1, t=0..1, color=green);

 

I wonder if it is possible to automatically obtain the well-known  binomial theorem  for an arbitrary integer and a positive exponent  n  in Maple. The expansion  (1)  below  I want to get in Maple automatically. But all my attempts were unsuccessful:
 

restart;

(a+b)^n=Sum(binomial(n,k)*a^(n-k)*b^k, k=0..n);  # The binomial theorem

(a+b)^n = Sum(binomial(n, k)*a^(n-k)*b^k, k = 0 .. n)

(1)

expand((a+b)^n) assuming n::posint;
convert((a+b)^n, Sum) assuming n::posint;
convert((a+b)^n, polynom) assuming n::posint;
convert((a+b)^n, binomial) assuming n::posint;

 

(a+b)^n

 

(a+b)^n

 

(a+b)^n

 

(a+b)^n

(2)

 


 

Download binomial_theorem.mw

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