Section: 17 | Transforms |
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John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), CRC Press/Taylor & Francis, Boca Raton, FL.
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7 TRANSFORMS

7.1 FOURIER TRANSFORMS

For a piecewise continuous function F(x) over a finite interval 0xπ ; the finite Fourier cosine transform of F(x) is fc(n)=0πF(x)cosnxdx   (n=0,1,2,) If x ranges over the interval 0xL , the substitution x=πxL allows the use of this definition, also. The inverse transform is written. F̲(x)=1πfc(0)2πn=1xfc(n)cosnx   (0<x<π) where F(x)=F(x+)+F(x)2 . Note that F(x+)=F(x)=F(x) at points of continuity. The formula fc(2)(n)=0πF(x)cosnxdx=n2fc(n)F(0)+(1)nF(π) makes the finite Fourier cosine transform useful in certain boundary value problems. Analogously, the finite Fourier sine transform of F(x) is fs(n)=0πF(x)sinnxdx   (n=1,2,3,) and F̲(x)=2πn=1fs(n)sinnx   (0<x<π) Corresponding to equation (33) we have fs(2)(n)=0πF(x)sinnxdx=n2fs(n)nF(0)n(1)nF(π) If F(x) is defined for x0 and is piecewise continuous over any finite interval, and if 0xF(x)dx is absolutely convergent, then fc(α)=2π0xF(x)cos(αx)dx is the Fourier cosine transform of F(x) . Furthermore, F̲(x)=2π0xfc(α)cos(αx)dα. If limxdnF/dxn=0 , then an important property of the Fourier cosine transform is fc(2r)(α)=2π0x(d2rFdx2r)cos(αx)dx=2πn=0r1(1)na2r2n1α2n+(1)rα2rfc(α) where limxdrF/dxr=ar, which is often useful. Under the same conditions, define the Fourier sine transform of F(x) as follows fs(α)=2π0xF(x)sin(αx)dx with F̲(x)=2π0xfs(α)sin(αx)dα Corresponding to (34) we have fs(2r)(α)=2π0(d2rFdx2rsin(αx))dx=2πn=1r(1)nα2n1a2r2n+(1)r1α2rfs(α) Similarly, if F(x) is defined for <x< , and if F(x)dx is absolutely convergent, then f(α)=12πF(x)eiaxdx is the Fourier transform of F(x) , and F̲(x)=12πf(α)eiaxdα Also, if   lim|x||dnFdxn|=0   (n=1,2,,r1)   then f(r)(α)=12πF(r)(x)eiαxdx=(iα)rf(α)

7.2 TABLE OF FOURIER COSINE TRANSFORMS

F(ω)=fc(f)(ω)=2π0f(x)cos(ωx)dx , ω>0.

No.f(x) F(ω)
1. {10<x<a0x>a 2πsinaωω
2. xp1(0<p<1) 2πΓ(p)ωpcospπ2
3. {cosx0<x<a0x>a 12π(sin[a(1ω)]1ω+sin[a(1+ω)]1+ω)
4. ex 2π11+ω2
5. ex2/2 eω2/2
6. cosx22 cos(ω22π4)
7. sinx22 cos(ω22+π4)

7.3 TABLE OF FINITE COSINE TRANSFORMS

fc(n)=0πF(x)cosnxdx , for n=0,1,2,.

No.fc(n) F(x)
1. (1)nfc(n) F(πx)
2. {πn=00n=1,2, 1
3. {0n=02nsinnπ2n=1,2, {1for0<x<π/21forπ/2<x<π
4. {π22n=0(1)n1/n2n=1,2 x
5. {π26n=0(1)n1/n2n=1,2 x22π
6. (1)necπ1n2+c2 1cecx
7. kn2k2[(1)ncosπk1] with k0,1,2, sinkx
8. {0m=1,2,(1)n+m1n2m2m1,2 1msinmx
9. 1n2k2 with k0,1,2, cosk(πx)ksinkπ
10. {π/2whenn=m0whennm cosmx      (m=1,2,)

7.4 TABLE OF FOURIER SINE TRANSFORMS

F(ω)=Fs(f)(ω)=2π0f(x)sin(ωx)dx,   ω>0.

No.f(x) F(ω)
1. {10<x<a0x>a 2π1cosωaω
2. xp1(0<p<1) 2πΓ(p)ωpsinpπ2
3. {sinx0<x<a0x>a 12π(sin[a(1ω)]1ωsin[a(1+ω)]1+ω)
4. ex 2πω1+ω2
5. xex2/2 ωeω2/2

7.5 TABLE OF FINITE SINE TRANSFORMS

fs(n)=0πF(x)sinnxdx , for n=1,2, .

No.fs(n) F(x)
1. (1)n+1fs(n) F(πx)
2. 1/n πx/π
3. (1)n+1/n x/π
4. 1(1)n/n 1
5. 2n2sinnπ2 {xwhen0<x<π/2πxwhenπ/2<x<π
6. (1)n+1/n3 x(π2x2)/6π
7. 1(1)n/n3 x(πx)/2
8. π2(1)n1n2[1(1)n]n3 x2
9. nn2+c2[1(1)necπ] ecx
10. nn2+c2 sinhc(πx)sinhcπ
11. nn2k2 with k0,1,2, sink(πx)sinkπ
12. {π/2whenn=m0whennm,m=1,2, sinmx
13. nn2k2[1(1)ncoskπ] with k1,2, (0 if n=k ) coskx
14. bnn with |b|1 2πarctanbsinx1bcosx
15. 1(1)nnbn with |b|1 2πarctan2bsinx1b2

7.6 TABLE OF FOURIER TRANSFORMS

F(ω)=F(f)(ω)=12πf(x)eiωxdx

No.f(x) F(ω)
1. δ(x) 1/2π
2. δ(xτ) eiωτ/2π
3. δ(n)(x) (iω)n/2π
4. H(x)={1x>00x<0 1iω2π+π2δ(ω)
5. sgn(x)={1x>01x<0 2π1iω
6. {1|x|<a1|x|>a 2πsinaωω
7. {eΩt|x|<a0|x|>a 2πsina(Ω+ω)Ω+ω
8. ea|x|a>0 2πaa2+ω2
9. sinaxx {π2|ω|<a0|ω|>a
10. {eiaxp<x<q0x<p,x>q i2πeip(ω+a)eiq(ω+a)/ω+a
11. {ecx+iaxx>00x<0(c>0) i2π(ω+a+ic)
12. epx2Rep>0 12peω2/4p
13. cospx2 12pcos(ω24pπ4)
14. sinpx2 12pcos(ω24p+π4)
15. |x|p(0<p<1) 2πΓ(1p)sinpπ2|ω|1p
16. ea|x|/|x| a2+ω2+aω2+a2
17. coshaxcoshπx(π<a<π) 2πcosa2coshω2cosa+coshω
18. sinhaxsinhπx(π<a<π) 12πsinacosa+coshω
19. {1a2x2|x|<a0|x|>a π2J0(aω)
20. sin[ba2+x2]a2+x2 {0|ω|>bπ2J0(ab2ω2)|ω|<b

7.7 TABLE OF FUNCTIONAL RELATIONS FOR FOURIER TRANSFORMS

F(ω)=F(f)(ω)=12πf(x)eiωxdx

No.f(x) F(ω)
1. ag(x)+bh(x) aG(ω)+bH(ω)
2. f(ax)a0,Ima=0 1|a|F(ωa)
3. f(x) F(ω)
4. f(x)̲ F(ω)̲
5. f(xτ)Imτ=0 eiωτF(ω)
6. eiΩxf(x)ImΩ=0 F(ω+Ω)
7. F(x) f(ω)
8. dndxnf(x) (iω)nF(ω)
9. (ix)nf(x) dndωnF(ω)

7.8 TABLE OF MULTIDIMENSIONAL FOURIER TRANSFORMS

F(u)=(2π)n/2nf(x)ei(xu)dx

No.f(x) F(u)
Two dimensions: let x=(x,y) and u=(u,v).
1. f(ax,by) 1|ab|F(ua,vb)
2. f(xa,yb) ei(au+bv)F(u,v)
3. ei(ax+by)f(x,y) F(u+a,v+b)
4. F(x,y) (2π)2F(u,v)
5. δ(xa)δ(yb) 12πei(au+bv)
6. ex2/4ay2/4ba,b>0 2abeau2bv2
7. {1|x|<a,|y|<b0otherwise(rectangle) 2sinausinbvπuv
8. {1|x|<a0otherwise(strip) 2sinauπuvδ(v)
9. {1x2+y2<a20otherwise(circle) aJ1(au2+v2)u2+v2
Three dimensions: let x=(x,y,z) and u=(u,v,w).
10. δ(xa)δ(yb)δ(zc) 1(2π)3/2ei(au+bv+cw)
11. ex2/4ay2/4bz2/4ca,b,c>0 23/2abceau2bv2cw2
12. {1|x|<a,|y|<b,|z|<c0otherwise(box) (2π)3/2sinausinbvsincwuvw
13. {1x2+y2+z2<a20otherwise(ball) sinaρaρcosaρ2πρ3      ρ2=u2+v2+w2

7.9 TABLE OF LAPLACE TRANSFORMS

F(s)=L(f)(s)=0f(t)estdt.

No.f(t) F(s)
1. δ(t),delta function 1
2. H(t),unit step function or Heaviside function 1/s
3. t 1/s2
4. tn1(n1)! 1/sn(n=1,2,)
5. 1/πt 1/s
6. 2t/π s3/2
7. 2ntn1/2π(2n1)!! s(n+1/2)(n=1,2,)
8. tk1 Γ(k)sk(k>0)
9. eat 1sa
10. teat 1(sa)2
11. 1(n1)!tn1eat 1(sa)n(n=1,2,)
12. tk1eat Γ(k)(sa)k(k>0)
13. 1ab(eatebt) 1(sa)(sb)(ab)
14. 1ab(aeatbebt) s(sa)(sb)(ab)
15. (bc)eat+(ca)ebt+(ab)ect(ab)(bc)(ca) 1(sa)(sb)(sc)(a,b,cdistinct)
16. 1asinat 1s2+a2
17. cosat ss2+a2
18. 1asinhat 1s2a2
19. coshat ss2a2
20. 1a2(1cosat) 1s(s2+a2)
21. 1a3(atsinat) 1s2(s2+a2)
22. 12a3(sinatatcosat) 1(s2+a2)2
23. t2asinat s(s2+a2)2
24. 12a(sinat+atcosat) s2(s2+a2)2
25. tcosat s2a2(s2+a2)2
26. cosatcosbtb2a2 s(s2+a2)(s2+b2)(a2b2)
27. 1beatsinbt 1(sa)2+b2
28. eatcosbt sa(sa)2+b2
29. sinatcoshatcosatsinhat 4a3s4+4a4
30. 12a2sinatsinhat ss4+4a4
31. 12a3(sinhatsinat) 1s4a4
32. 12a2(coshatcosat) ss4a4
33. (1+a2t2)sinatcosat 8a3s2(s2+a2)3
34. etn!dndtn(tnet) 1s(s1s)n
35. 1πteat(1+2at) s(sa)3/2
36. 12πt3(ebteat) sasb
37. 1πtaea2terfc(at) 1s+a
38. 1πt+aea2terf(at) ssa2
39. 1aea2terf(at) 1s(sa2)
40. {0when0<t<k1whent>k ekss
41. {0when0<t<ktkwhent>k ekss2
42. {0when0<t<k(tk)p1Γ(p)when t>k ekssp   (p > 0)
43. {1when0<t<k0whent>k 1ekss
44. |sinat| as2+a2cothπs2a
45. J0(2at) 1sea/s
46. 1πtcos2at 1sea/s
47. 1πtcosh2at 1sea/s
48. a2πtea2/4t eas(a>0)
49. 1πtea2/4t 1seas(a0)
50. Γ(1)logt 1slogs
51. tk1[Γ(k)|Γ(k)|2logtΓ(k)] 1sklogs(k>0)
52. eat[logaEi(at)] logssa(a>0)
53. 1t(ebteat) logsasb
54. 2t(1cosat) logs2+a2s2
55. 2t(1coshat) logs2a2s2
56. 1tsinat arctanas
57. 1aπet2/4a2 ea2s2erfc(as)(a>0)
58. erf(t2a) 1sea2s2erfc(as)(a>0)
59. aπt(t+a) easerfc(as)(a>0)
60. 1πtsin(2at) erf(as)

7.10 TABLE OF FUNCTIONAL RELATIONS FOR LAPLACE TRANSFORMS

F(s)=L(f)(s)=0f(t)estdt.

No.f(t) F(s)
1. af(t)+bg(t) aF(s)+bG(s)
2. f(t) sF(s)F(0+)
3. f(t) s2F(s)sF(0+)F(0+)
4. f(n)(t) snF(s)k=0n1sn1kF(k)(0+)
5. 0tf(τ)dτ 1sF(s)
6. 0t0τf(u)dudτ 1s2F(s)
7. 0tf1(tτ)f2(τ)dτ=f1*f2 F1(s)F2(s)
8. tf(t) F(s)
9. tnf(t) (1)nF(n)(s)
10. 1tf(t) sF(z)dz
11. eatf(t) F(sa)
12. f(tb)withf(t)=0fort<0 ebsF(s)
13. 1cf(tc) F(cs)
14. 1cebt/cf(tc) F(csb)
15. f(t+a)=f(t) 0aestf(t)dt/1eas
16. f(t+a)=f(t) 0aestf(t)dt/1+eas
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