Your post seems to consist of image tags pointing to files that you have on your own computer (e.g.
C:/Users/JeanPhi/AppData/Local/Temp/moz-screenshot-29.png), but which we have no access to.

If you want to upload files, you can use the green up-arrow button.  But the best thing to do is simply type your question, including any equations or Maple code, in ordinary text.  Use the Preview comment button to see what it your posting will look like before you post it.  If you see html code such as <p> and &nbsp;, click on "Input format" and change it to "Filtered HTML" instead of "Plain text".

Your post seems to consist of image tags pointing to files that you have on your own computer (e.g.
C:/Users/JeanPhi/AppData/Local/Temp/moz-screenshot-29.png), but which we have no access to.

If you want to upload files, you can use the green up-arrow button.  But the best thing to do is simply type your question, including any equations or Maple code, in ordinary text.  Use the Preview comment button to see what it your posting will look like before you post it.  If you see html code such as <p> and &nbsp;, click on "Input format" and change it to "Filtered HTML" instead of "Plain text".

One thing to note about Eigenvalues: if your B has rational entries, it will look for an exact expression for the eigenvalues by solving the characteristic polynomial.  This may not be what you want.  If you want numerical approximations for the eigenvalues, you should either make sure that B contains at least one float or use

> Eigenvalues(evalf(M));

One thing to note about Eigenvalues: if your B has rational entries, it will look for an exact expression for the eigenvalues by solving the characteristic polynomial.  This may not be what you want.  If you want numerical approximations for the eigenvalues, you should either make sure that B contains at least one float or use

> Eigenvalues(evalf(M));

I don't think Maple "guesses" or ":chooses" a vertical range per se.  It calculates points using its adaptive plotting algorithm (see the section on adaptive in the help page ?plot,options). The vertical range is then taken to fit all the values that have been calculated.

I don't think Maple "guesses" or ":chooses" a vertical range per se.  It calculates points using its adaptive plotting algorithm (see the section on adaptive in the help page ?plot,options). The vertical range is then taken to fit all the values that have been calculated.

This is two separate loops, not a nested loop.  Read again what I wrote.

This is two separate loops, not a nested loop.  Read again what I wrote.

Another example of why it's best to avoid 2D math mode.

Another example of why it's best to avoid 2D math mode.

I think you mean this.

> S:=simplify(sum(x^k/product(2*j+1,j=0..k),k=0..infinity));
S := 1/2*2^(1/2)*exp(1/2*x)*Pi^(1/2)*erf(1/2*2^(1/2)*x^(1/2))/x^(1/2)

> series(S, x);
1+1/3*x+1/15*x^2+1/105*x^3+1/945*x^4+1/10395*x^5+O(x^(11/2))

I think you mean this.

> S:=simplify(sum(x^k/product(2*j+1,j=0..k),k=0..infinity));
S := 1/2*2^(1/2)*exp(1/2*x)*Pi^(1/2)*erf(1/2*2^(1/2)*x^(1/2))/x^(1/2)

> series(S, x);
1+1/3*x+1/15*x^2+1/105*x^3+1/945*x^4+1/10395*x^5+O(x^(11/2))

The region between the two circles |z-1|=1 and |z-2|=2 corresponds, by the conformal mapping w = 1/z, to the region between the two straight lines Re(w) = 1/4 and Re(w) = 1/2.

> with(plots): 
  conformal(1/w, w=1/4 - 10*I .. 1/2 + 10*I, scaling=constrained, numxy=[30,1000],
     grid=[5,400],colour=[red,blue]);

The region between the two circles |z-1|=1 and |z-2|=2 corresponds, by the conformal mapping w = 1/z, to the region between the two straight lines Re(w) = 1/4 and Re(w) = 1/2.

> with(plots): 
  conformal(1/w, w=1/4 - 10*I .. 1/2 + 10*I, scaling=constrained, numxy=[30,1000],
     grid=[5,400],colour=[red,blue]);

The link should have been

www.mapleprimes.com/files/9288_integral output.pdf

(Maple Primes seems to have had trouble with the space in the file name).

The integral is

> int(exp(-I*k*sqrt(r^2-2*a*r*sin(theta)*cos(phi)+a^2)),phi=0..2*Pi);

My best guess is that there is no closed form for this.  You could e.g. express the result as a series in powers of a:

> series(%,a) assuming r > 0;

series(2*exp(-I*k*r)*Pi+(-1/2*k*Pi*(I+k*r-k*r*cos(theta)^2+cos(theta)^2*I)*exp(-I*k*r)/r)*a^2+(-1/32*k*Pi*(-k^3*r^3*cos(theta)^4-2*I*k^2*r^2+15*k*r*cos(theta)^4+2*k^3*r^3*cos(theta)^2+I+6*I*cos(theta)^2-6*k*r*cos(theta)^2-4*I*k^2*r^2*cos(theta)^2+6*I*k^2*r^2*cos(theta)^4-15*I*cos(theta)^4-k^3*r^3-k*r)*exp(-I*k*r)/r^3)*a^4+O(a^6),a,6)