https://en.wikipedia.org/wiki/Euler%27s_formula

For x=0 the term is not defined

@rcorless certainly you meant @vv , not me ...   :-)

A download does not work for me.

What is the filename? Have you tried a name having only ASCII letters and numbers? Have you contacted the technical support by mail?

NB: I am using Win, not Mac

Please post code instead of images.

@BasSPe 

'u_tilde*exp(I*omega*t)/2/Pi';
'eval(%, omega=-omega) + %';
simplify(%) assuming 0<= omega:
evalc(Re(%))  assuming 0<= omega:
#map(identify, %):
vRe:=unapply(%, omega, t);

Now one wants to integrate vRe from 0 to 40 (instead of infinity), giving a function it t.

vRe(omega, 1); plot(%, omega=14.5 .. 14.6); # (t=1 as an example)

That plotting shows something which probably needs a Cauchy Principal Value and it is no clear (for me) why summation is good enough (omega = 10 is ok, it cancels out)

I do not quite understand why one can cut off at 0 and 40 and why that summing is a reasonable or rough approximation.

The integrand has singularities - the denominator only depends on omega and has zeroes, omega = 10 gives a pole for example.

You should post it as code and care for correct brackets

I do not quite understand your sheet, but having Sum(f(n), n= a .. b) and setting g:= k -> f(5*k) then Sum(g(n), n= a .. b) might be what you are looking for. So write your summands as function f.

Likewise and not caring for the discontinuity:

G:=w -> Int(eval(A1+A2,omega=w),theta=0..2*Pi, method = _d01ajc, epsilon=1e-3);

Digits:=15;
plot(G, 0 .. 50);

@mthkvv 

My point is: Maple uses the conventions (or I get you wrong), I do not want to discuss conventions

@markweitzman 

Here it may be simple/elegant - but what about running up to Pi + Pi/4 or Pi + Pi/2 ?

@vv 

Maple uses the conventions in Abramowitz & Stegun or DLMF, https://dlmf.nist.gov/

I am not aware of a commitment though, say through Maple's documentation

@Axel Vogt 

Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/prove.mw .
 

Download prove.mw