COMPLEX ANALYSIS: Mathematica 4.1 Notebooks
(c) John H. Mathews, and
Russell W. Howell, 2002
Complimentary software to accompany our textbook:

COMPLEX ANALYSIS: for Mathematics and Engineering, 
4th Edition,  2001,  ISBN: 0-7637-1425-9
Jones & Bartlett Publishers, Inc.  
40 Tall Pine Drive, Sudbury, MA 01776  
Tele.  (800) 832-0034, FAX:  (508)  443-8000  
E-mail:  mkt@jbpub.com  
Internet:  http://www.jbpub.com/  
This free software is compliments of the authors.
John H. Mathews,   mathews@fullerton.edu 
Russell W. Howell,  howell@westmont.edu

CHAPTER 11  FOURIER SERIES and the LAPLACE TRANSFORM

a = A = b = B = c = d = s = S = t = T = u = U = v = V = w = W = x = X = y = Y = z = Z = 
                                        0; 
Clear[a,a1,a2,a3,A,A1,A2,b,b1,b2,B1,B2,B,c,d,EqA,EqB,f,f1,f2,f12,F,F1,F2,F12,F3,g,G,h,H,i,int,j,k,L,Ly,L1,L2,Lf,LDf,ℒf,ℒDf,n,PQ,PR,r,res1,res2,s,solset,s1,s2,s3,S,S9,t,table,τ,T,θ,u,U,U1,U2,U7,v,v1,v2,v3,v4,val,V,w,w1,w2,W,x,X,y,y0,y1,Y,Yi,z,z1,z2,Z]; 
Needs["Graphics`Colors`"]; 
Needs["Calculus`FourierTransform`"];

Section 11.1	Fourier Series

Section 11.2	The Dirichlet Problem for the Unit Disk

Section 11.3	Vibrations in Mechanical Systems

Section 11.4  The Fourier Transform

In this section we develop the complex Fourier transform of the function U(t).

Theorem 11.9 (Fourier Transform), Page 508. Let U(t)and  U'(t) be piecewise continuous, and  \(∫\_\(-∞\)\%∞\(\(|\)\(U \((t)\)\)\(|\)\(dt\  < \ M\)\)\),  for some positive constant  M.  The Fourier transform  F (w)  of  U (t)  is defined as  

		\(F \((w)\) = \(1\/\(2  π\)\) \(∫\_\(-∞\)\%∞\ 
            U \((t)\) \(e\^\(\(-i\)\ w\ t\)\) dt\)\).  

At points of continuity,  U (t)  has the integral representation  

		\(U \((t)\) = ∫\_\(-∞\)\%∞\ 
          F \((w)\) \(e\^\(\(-i\)\ w\ t\)\) dw\),  

and at a point  t = a  of discontinuity of  U (t),  this integral converges to  \(\(U \((\(a\^-\))\)\  + \ U \((\(a\^+\))\)\)\/2\).

Proof of Theorem 11.9, see text Page 508.

Example 11.5,  Page 510.  Find the Fourier transform of  U (t) = (e^(-|t|)).

Solution 11.5.

Enter the integrand  U (t) = (e^(-|t|)) and perform integration.

Clear[F, F1, F2, F3, t, U, U1, U2, w, w1, 
        w2]; 
U[t_]  = (E^(-Abs[t])); 
U1[t_]  = (E^t); 
U2[t_]  = (E^(-t)); 
w1[
          t_]  = ∫U1[
              t] (E^(-I w t)) \[DifferentialD]t; 
w2[
          t_]  = ∫U2[
              t] (E^(-I w t)) \[DifferentialD]t; 
Print["U[t] = ", U[t]]; 
Print[""]; 
\(\(Print[\*""\!\(U\_1\)[t] = "", U1[t]];\)\ \)
\(\(Print[\*""\!\(U\_2\)[t] = "", U2[t]];\)\ \)
\(\(Print[\*""(Subscript[w, 1])[t] = ∫"", 
        U1[t], \*""(E^(-Iwt)) \[DifferentialD]t = "", w1[t]];\)\ \)
\(\(Print[\*""(Subscript[w, 2])[t] = ∫"", 
        U2[t], \*""(E^(-Iwt)) \[DifferentialD]t = "", w2[t]];\)\ \)

Because of the exponential order  (Subscript[w, 1])[-∞] = 0  and  (Subscript[w, 2])[∞] = 0  and the Fourier transform is given by the following computation:

F[w_]  =  
        Together[((w1[0] - w2[0])/(2  π))]; 
Print["U[t] = ", U[t]]; 
Print[""]; 
\(\(Print[\*""F[w] = (1/(2  π))((Subscript[w, 1])[0] - (Subscript[w, 2])[0])""];\)\ \)
\(\(Print[\*""\!\(w\_1\)[0] = "", 
        w1[0]];\)\ \)
\(\(Print[\*""\!\(w\_2\)[0] = "", 
        w2[0]];\)\ \)
Print[""]; 
Print["F[w] = ", (1/(2  π)), "((", w1[0], ") - (",  
        w2[0], "))"]; 
Print["F[w] = ", F[w]];

Or we can let Mathematica  work with some very complicated improper integrals.

val1  =  
        Integrate[
          U1[t] (E^(-I w t)), {t, -∞, 0}, Assumptions -> {Im[w] == 0}]; 
val2  =  
        Integrate[
          U2[t] (E^(-I w t)), {t, 
            0, ∞}, 
          Assumptions -> {Im[w] == 0}]; 
F1  =  ((val1  +  val2)/(2  π)); 
F2  =  Together[((val1  +  val2)/(2  π))]; 
F3  =  
        ExpandAll[Together[((val1  +  val2)/(2  π))]]; 
F[w_]  =  
        Factor[ExpandAll[
            Together[((val1  +  val2)/(2  π))]]]; 
Print["U[t] = ", U[t]]; 
Print[""]; 
\(\(Print[\*""(Subscript[v, 1]) = (∫_(-∞)^0)"", 
        U1[t], \*""(E^(-Iwt)) \[DifferentialD]t = "", val1];\)\ \)
\(\(Print[\*""(Subscript[v, 2]) = (∫_0^∞)"", 
        U2[t], \*""(E^(-Iwt)) \[DifferentialD]t = "", val2];\)\ \)
Print[""]; 
\(\(Print[\*""F[w] = (1/(2  π))((Subscript[v, 1]) + (Subscript[v, 2])) = "", 1\/\(2  π\), "((", val1, ") - (", \ 
        val2, "))"];\)\ \)
Print["F[w] = ", F1, " = ", F2, " = ", F3]; 
Print["F[w] = ", F[w]];

Aside. Let us use Mathematica  to find the FourierTransform .

Fm[w_]  =  
        FourierTransform[U[t],  t,  ω]; 
Print[" U[t] = ", U[t]]; 
Print[""]; 
\(\(Print["Mathematica's version of the Fourier Transform is"];\)\ \)
\(\(Print[\*""\!\(F\_m\)[w] = "", 
        Fm[w]];\)\ \)

Remark. Mathematica's  FourierTransform package does not use the same coefficient involving  π  as is commonly used in complex analysis.

Example 11.6,  Page 510.  Find the Fourier transform of  U (t) = (1/(1 + (t^2))).

Solution 11.6.

Use the result of Example 11.5 and the symmetry property on page 421 to find the Fourier transform.
Symmetry property.  If  ℱ(U (t))=F (w)  then  ℱ(F (t))=(1/(2  π)) U (-w).

Clear[f, F, t, u, U, w]; 
u[t_]  = (E^(-Abs[t])); 
f[w_] = (1/(π (1 + (w^2)))); 
U[t_]  = π f[t]; 
F[w_]  = π (1/(2  π)) 
          u[-w]; 
\(\(Print["The known function u[t] and Fourier transform f[w]"];\)\ \)
Print["u[t] = ", u[t]]; 
Print["f[w] = ", f[w]]; 
U[t_]  =  f[t]; 
F[w_]  =  (1/(2  π)) 
          u[-w]; 
Print[""]; 
Print["Apply the symmetry property"]; 
Print["U[t] = ", U[t]]; 
Print["F[w] = ", F[w]]; 
Print[""]; 
U[t_]  =  π f[t]; 
F[w_]  = π (1/(2  π)) 
          u[-w]; 
Print["Multiply by π for the desired result"]; 
Print["U[t] = ", U[t]]; 
Print["F[w] = ", F[w]];

Aside. Let us use Mathematica  to find the FourierTransform .

Fm[w_]  =  
        FourierTransform[U[t],  t,  ω]; 
Print[" U[t] = ", U[t]]; 
Print[""]; 
\(\(Print["Mathematica's version of the Fourier Transform is"];\)\ \)
\(\(Print[\*""\!\(F\_m\)[w] = "", 
        Fm[w]];\)\ \)

Remark. Mathematica's  FourierTransform package does not use the same coefficient involving  π  as is commonly used in complex analysis.

Library Research Experience for Undergraduates

Project I. Write a report on the Fourier transform.

1. Bleick, W. E., ''Fourier Analysis of Engine Unbalance by Contour Integration,'' Am. Math. M., Vol. 63, No. 7. (Aug. - Sep., 1956), pp. 466-472., Jstor.   

2. Boas, R. P., (1962), ''Inversion of Fourier and Laplace Transforms,'' Am. Math. M., Vol. 69, No. 10. (Dec., 1962), pp. 955-960, Jstor.   

3. Fisher, Stephen D., (1990), Complex Variables, 2nd ed., Monterey, Calif., Wadsworth & Brooks/Cole.

4. Henrici, Peter, ''Fast Fourier Methods in Computational Complex Analysis,'' SIAM Review, Vol. 21, No. 4. (Oct., 1979), pp. 481-527., Jstor.   

5. Kerley, Lyndell M. and Jeff R. Knisley, (1993), ''Complex Vectors and Image Identification,'' Coll. Math. J., V. 24, No. 2, pp. 166-174.

6. Priestley, Hilary A., (1985), Introduction to Complex Analysis, N.Y., Oxford University Press.

7. Reid, Walter P., ''From Fourier Series to Fourier Integral (in Classroom Notes),'' Am. Math. M., Vol. 64, No. 9. (Nov., 1957), pp. 663-664., Jstor.   

8. Saff, E. B. and A. D. Snider, (1993), Complex Analysis for Math., Science and Engineering, Englewood Cliffs, N.J., Prentice Hall.

9. Schoenberg, I. J., ''The Finite Fourier Series and Elementary Geometry Am. Math. M., Vol. 57, No. 6. (Jun. - Jul., 1950), pp. 390-404, Jstor.   

10. Solomon, Frederick, (1979), ''Tomography:\ Three Dimensional Image Reconstruction,'' UMAP, Unit 318, pp. 1-19, COMAP, Inc. Lexington, MA 02420 USA. COMAP

Section 11.4  Exercises for The Fourier Transform
See textbook page 510.

Section 11.5	The Laplace Transform

Section 11.6	Laplace Transforms of Derivatives and Integrals

Section 11.7	Shifting Theorems and the Step Function

Section 11.8	Multiplication and Division by  t

Section 11.9	Inverting the Laplace Transform

Section 11.10	Convolution

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