# Harj. 10
# 22-23.11.01
Warning, the name changecoords has been redefined

# 1.
> cn:=1/T*(Int(-exp(-I*n*omega*t),t=-1..0)+Int(exp(-I*n*omega*t),t=0..1)
> );

              0                            1
             /                            /
            |                            |
            |   -exp(-I n omega t) dt +  |   exp(-I n omega t) dt
            |                            |
           /                            /
             -1                           0
     cn := ------------------------------------------------------
                                     T

> T:=2: omega:=2*Pi/T;

                             omega := Pi

> value(cn);cn:=evalc(%);

         1/2 I (-1 + exp(I n Pi))   1/2 I (exp(-I n Pi) - 1)
         ------------------------ + ------------------------
                   n Pi                       n Pi


                        2 I (- 1/2 + 1/2 cos(n Pi))
                  cn := ---------------------------
                                   n Pi

> cmiinus_n:=1/T*(Int(-exp(-I*(-n)*omega*t),t=-1..0)+Int(exp(-I*(-n)*ome
> ga*t),t=0..1));

                     0                            1
                    /                            /
                   |                            |
 cmiinus_n := 1/2  |   -exp(I n Pi t) dt + 1/2  |   exp(I n Pi t) dt
                   |                            |
                  /                            /
                    -1                           0

> value(cmiinus_n);cmiinus_n:=evalc(%);
> 

        -1/2 I (exp(-I n Pi) - 1)   -1/2 I (-1 + exp(I n Pi))
        ------------------------- + -------------------------
                  n Pi                        n Pi


                            2 I (- 1/2 cos(n Pi) + 1/2)
               cmiinus_n := ---------------------------
                                       n Pi

> cn*exp(I*n*omega*t)+cmiinus_n*exp(-I*n*omega*t);evalc(%);cnn:=expand(%
> );

  2 I (- 1/2 + 1/2 cos(n Pi)) exp(I n Pi t)
  -----------------------------------------
                    n Pi

           2 I (- 1/2 cos(n Pi) + 1/2) exp(-I n Pi t)
         + ------------------------------------------
                              n Pi


    (cos(n Pi) - 1) sin(Pi n t)   (-cos(n Pi) + 1) sin(Pi n t)
  - --------------------------- + ---------------------------- + I
               n Pi                           n Pi

        /(cos(n Pi) - 1) cos(Pi n t)   (-cos(n Pi) + 1) cos(Pi n t)\
        |--------------------------- + ----------------------------|
        \           n Pi                           n Pi            /


                     sin(Pi n t) cos(n Pi)   2 sin(Pi n t)
           cnn := -2 --------------------- + -------------
                             n Pi                n Pi

> collect(cnn,sin(Pi*n*t));factor(%);

                  /   cos(n Pi)    2  \
                  |-2 --------- + ----| sin(Pi n t)
                  \     n Pi      n Pi/


                       (cos(n Pi) - 1) sin(Pi n t)
                    -2 ---------------------------
                                  n Pi

> factor((-2*cos(n*Pi)/(n*Pi)+2/(n*Pi)));

                              cos(n Pi) - 1
                           -2 -------------
                                  n Pi

> 
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