Q1a. HeavisideTheta x. Plot f, x, Pi, Pi. Simplify, n Integers

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1 2 M2 Fourier Series answers in Mathematica Note the function HeavisideTheta is for x>0 and 0 for x<0 and is handy for making the kind of function we need in a way mathematica knows how to integrate I have plotted the Fourier series (at least afew terms) even though this was not asked for as this gives you an idea how accurate the series is. Even if you don t want to use Mathematica to check your integration please have a look at the graphs Qa f HeavisideTheta x HeavisideTheta x Plot f, x, Pi, Pi a0 Pi Integrate f, x, Pi, Pi an Pi Integrate f Cos n Pi x Pi, x, Pi, Pi Sin 0 bn Pi Integrate f Sin n Pi x Pi, x, Pi, Pi Cos n

2 2 FourierAnswers.nb Plot a0 2 Sum an Cos n x bn Sin n x, n,, 20, x, Pi, Pi Qb f HeavisideTheta x HeavisideTheta Pi 2 x HeavisideTheta x HeavisideTheta x 2 Plot f, x, Pi, Pi a0 Pi Integrate f, x, Pi, Pi 2 an Pi Integrate f Cos n Pi x Pi, x, Pi, Pi Sin n 2 Sin n 2 bn Pi Integrate f Sin n Pi x Pi, x, Pi, Pi 2 Sin n 4 2

3 FourierAnswers.nb 3 Plot a0 2 Sum an Cos n x bn Sin n x, n,, 20, x, Pi, Pi Qc f 3 HeavisideTheta x 3 HeavisideTheta x Plot f, x, 5, a0 5 Integrate f, x, 5, 5 3 an 5 Integrate f Cos n Pi x 5, x, 5, 5 3 Sin 0 bn 5 Integrate f Sin n Pi x 5, x, 5, 5 3 Cos 3 n

4 4 FourierAnswers.nb Plot a0 2 Sum an Cos n Pi x 5 bn Sin n Pi x 5, n,, 20, x, 5, Qd f x x a0 2 Integrate f, x, 0, an 2 Integrate f Cos 2 n Pi x, x, 0, Cos 2 2 Sin 2 2 n bn 2 Integrate f Sin 2 n Pi x, x, 0, 2 Cos 2 Sin 2 2 n 2 2 Plot a0 2 Sum an Cos 2 n Pi x bn Sin 2 n Pi x, n,, 20, x, 0,.0 Qe

5 FourierAnswers.nb 5 f x^2 x 2 a0 2 Integrate f, x, 2, 2 6 an 2 Integrate f Cos 2 n Pi x, x, 2, 2 2 n 3 2 Cos 2 3 n2 2 Sin n n 2 2 bn 2 Integrate f Sin 2 n Pi x, x, 2, 2 0 Plot a0 2 Sum an Cos 2 n Pi x bn Sin 2 n Pi x, n,, 3, x, 2, Plot a0 2 Sum an Cos 2 n Pi x bn Sin 2 n Pi x, n,, 0, x,, Qf f Abs Sin w t Abs Sin t w

6 6 FourierAnswers.nb a0 Pi w Integrate f, t, Pi w, Pi w 4 w If w Reals,, Integrate Abs Sin t w, t, w w, w, Assumptions Im w 0 Im w 0 Simplify, w 0 4 an Pi w Integrate f Cos Pi n t Pi w, t, Pi w, Pi w 2 Cos w If w Reals,, w n 2 w Integrate Abs Sin t w Cos n t w, t, w, w, Assumptions Im w 0 Im w 0 Simplify, w 0 2 Cos n 2 bn Pi w Integrate f Sin Pi n t Pi w, t, Pi w, Pi w w If w Reals, 0, Integrate Abs Sin t w Sin n t w, t, w, w, Assumptions Im w 0 Im w 0 Simplify, w 0 0 w Plot 2 Cos Sum Cos n w x, n, 2, 0, x, Pi 2, Pi 2 n 2 Extra f Abs x Abs x

7 FourierAnswers.nb 7 a0 Integrate f, x,, an Integrate f Cos Pi n x, x,, 2 Cos Sin n 2 2 Plot a0 2 Sum an Cos n Pi x, n,, 0, x,,.0 Plot a0 2 Sum an Cos n Pi x, n,, 20, Sum 2 Pi ^n n Sin n Pi x, n,, 20, x,, Sum N n^2, n,, Sum N ^ n n, n,, Q2a f Sign x Out[5]= Sign x

8 8 FourierAnswers.nb In[7]:= Plot f, x, Pi, Pi Out[7]= It is odd In[9]:= Out[9]= In[]:= bn Pi Integrate f Sin n x, x, Pi, Pi 2 Cos Plot Sum bn Sin n x, n, 20, x, Pi, Pi.0 Out[]= Q2b In[2]:= Out[2]= f Abs x Abs x

9 FourierAnswers.nb 9 In[4]:= Plot f, x, Pi 2, Pi Out[4]= It is even In[29]:= Out[29]= In[32]:= an Simplify 2 Pi Integrate f Cos 2 n x, x, Pi 2, Pi 2 Cos Sin n 2 Plot Pi 4 Sum an Cos 2 n x, n, 30, x, Pi, Pi.5.0 Out[32]= Q3 In[57]:= fp x HeavisideTheta x x HeavisideTheta x Out[57]= In[54]:= Out[54]= x HeavisideTheta x x HeavisideTheta x fs x x In[55]:= Out[55]= fc Abs x Abs x

10 0 FourierAnswers.nb In[58]:= Plot fp, fs, fc, x,, Out[58]=.0 In[66]:= bpn Simplify 2 Integrate fp Sin 2 n Pi x, x, 0,, Element n, Integers Out[66]= In[65]:= Out[65]= In[77]:= Out[77]= In[80]:= bsn Simplify 2 2 Integrate fs Sin n Pi x, x,,, Element n, Integers 2 n acn Simplify 2 2 Integrate fc Cos n Pi x, x,,, Element n, Integers 2 n n 2 2 Plot 2 Sum bpn Sin 2 n Pi x, n,, 20, Sum bsn Sin n Pi x, n,, 20, 2 Sum acn Cos n Pi x, n,, 20, x,,.0 Out[80]=.0.0 Q4 In[83]:= Out[83]= f HeavisideTheta x HeavisideTheta x

11 FourierAnswers.nb In[85]:= Plot f, x, 0, 2.0 Out[85]= In[89]:= Out[89]= In[90]:= fc HeavisideTheta x HeavisideTheta x HeavisideTheta x HeavisideTheta x Plot fc, x, 2, 2 Out[90]= 2 In[93]:= fs HeavisideTheta x HeavisideTheta x ; Plot fs, x, 2, 2 Out[93]= 2 2 In[94]:= Out[94]= In[96]:= Out[96]= bpn Simplify 2 2 Integrate f Sin n Pi x, x, 0, 2, Element n, Integers n acn Simplify 2 4 Integrate fc Cos n Pi x 2, x, 2, 2, Element n, Integers 2 Sin n 2

12 2 FourierAnswers.nb In[97]:= bsn Simplify 2 4 Integrate fs Sin n Pi x 2, x, 2, 2, Element n, Integers Out[97]= In[09]:= 2 n Cos n 2 Plot 2 Sum bpn Sin n Pi x, n,, 0, 2 Sum acn Cos n Pi x 2, n,, 0, Sum bsn Sin n Pi x 2, n,, 0, x, 2, 2.0 Out[09]= Q5 In[36]:= Out[36]= In[37]:= Out[37]= In[38]:= Out[38]= In[50]:= Out[50]= In[40]:= Out[40]= 2 Cos. n, 2, 3, 4, 5, 6 4 4, 0, 3 4, 0, 5, 0 Integrate Sign x ^2, x, Pi, Pi 2 2 Cos Pi 4. n, 2, 3, 4, 5, 6, 0, 3, 0, 5, 0 Integrate Pi Sign x 4 ^2, x, Pi, Pi Pi 2 8 Sum Cos Pi 4 ^2, n,, Infinity Q6 In[4]:= Out[4]= In[46]:= Out[46]= n an n 2 2 n n 2 2 Integrate x^2, x, 2, 2 2

13 FourierAnswers.nb 3 In[49]:= Integrate Pi^2 x^2 2 ^2, x, 2, 2 2 Out[49]= In[44]:= Out[44]= 4 90 Sum n^4, n,, Infinity 4 90

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