中科大数理方程课件3

�, ��z � F �y (Fourier y ) �* � f(x)\�KT;ÆZ�q1EN� \�p�g2hS�U ∫ +∞ −∞ |f(x)|dx < +∞, ^�I$ fˆ(λ) = ∫ +∞ −∞ f(x)e−iλxdx 0 f(x)' Fouriery ,f(x)0 fˆ(λ)' Fourier�y ,[X0 fˆ(λ) = F [f(x)], f(x) = F−1[fˆ(λ)]. �'�� \�#�*9�\ f(x) 'qE*� f(x) = 1 2pi ∫ +∞ −∞ fˆ(λ)eiλxdλ 0(%/ 1. #% F [c1f1 + c2f2] = c1F [f1] + c2F [f2], 2. �)�% F [f(x)eiλ0x] = fˆ(λ− λ0) 3. ���! � f(±∞) = f ′(±∞) = · · · = f (n−1)(±∞) = 0, f (n)(x) ' Fourier P \�^ F [f (n)(x)] = (iλ)nfˆ(λ). ) ( F [f ′(x)] = iλfˆ(λ), F [f ′′(x)] = −λ2fˆ(λ). 4. ���� � f(x), g(x) \�p�g2hS�-PfSI$ f ∗ g(x) = ∫ +∞ −∞ f(x− ξ)g(ξ)dξ, ^ F [f ∗ g] = fˆ(λ)gˆ(λ). x.#���~ Fourier -"�+"y �* f(x)\ [0,+∞)��OT;ÆZ�qVEN� \ [0,+∞)�g2hS� 77 -"y � fˆs(λ) = Fs[f(x)] , ∫ +∞ 0 f(x) sinλxdx, +"y � fˆc(λ) = Fc[f(x)] , 2 pi ∫ +∞ 0 f(x) cos λxdx. �'�� 6e: P� f(x) = F−1s [fˆs(λ)] , ∫ +∞ 0 fˆs(λ) sin λxdλ, 6V: P� f(x) = F−1c [fˆc(λ)] , 2 pi ∫ +∞ 0 fˆc(λ) cos λxdλ. L �y Laplace y �* � f(t) = f(t)h(t), �m h(t) = { 1, t ≥ 0 0, t < 0 , p 0<$�^�I$ fˆ(p) = ∫ +∞ 0 f(t)e−ptdt 0 f(t)' Laplace y ,f(t)0 fˆ(p)' Laplace �y ,[X0 f¯(p) = L[f(t)], f(t) = L−1[f¯(p)]. �'�� # f(t) \ [0,+∞) ��OT;ÆZq1EN�^0j$_�B� [f(t)] ≤ keσ0t, σ0 ≥ 0, ^ f¯(p) \ Rep > σ0 �a7� f(t) = 1 2pii ∫ σ+i∞ σ−i∞ f¯(p)eptdp, σ > σ0. 0(%/ 1. #% L[c1f1 + c2f2] = c1L[f1] + c2L[f2]. 2. &|%/ L[f(t− z)] = e−pτL[f(t)], f(t− τ) = f(t− τ)h(t− τ). 3. ���! L[f (n)(t)] = pnL[f(t)]− pn−1f(+0)− pn−2f ′(+0) · · · − f (n−1)(+0). 78 4. ���� �* f ∗ g(t) = ∫ t 0 f(t− τ)g(τ)dτ, ^ L[f ∗ g] = L[f ] · L[g]. S: P5� a�/:7�'vJnoH75�rH RU5bÆW�'� /:7�'�i4+��i4+JQL4+�IhMRj aTbÆW�'QL4 +� S: P5.GS: P�2> s'/:Z% 0!$Z%��3��/ :7�'4+ 0�/:7�'4+ a�uM[oR6 P �X-a4+' a� �`rH(zR Fourier P (F �y ) J Laplace P (L �y ) a�/ :7�-a4+� ��{1 1. F℄v P' s�2d>-a4+ve P�&%2fI$'>I$|t '�/:7�'-a4+� 2. a�I$|t'�/:7�'-a4+� 3. 2�I$v6 P� �X�U2fI$� �$�3 � 3.1 5��=�'��$4+{ ∂u ∂t = a2 ∂2u ∂x2 + f(t, x), (t > 0,−∞ < x < +∞) u(0, x) = ϕ(x) Æ )J��ve P� bn −∞ < x < +∞, QF x sv F �y , X uˆ(t, λ) = F [u(t, x)] =∫ +∞ −∞ u(t, x)e−iλxdx. : 27�J��-℄v F ! P�&�/:7��i4+ { ∂uˆ ∂t = −a2λ2uˆ+ fˆ(t, λ) uˆ(0, λ) = ϕˆ(λ). )4�� a>I$� 79 7�.a0 uˆ(t, λ) = e−a 2λ2t [∫ t 0 fˆ(τ, λ)ea 2λ2τdτ + c ] !���-℄ uˆ(0, λ) = c = ϕˆ(λ) a& uˆ(t, λ) = ϕˆ(λ)e−a 2λ2t + ∫ t 0 fˆ(τ, λ)e−a 2λ2(t−τ)dτ )���v6 P�& u(t, x) = F−1[ϕˆ(λ)e−a 2λ2t] + F−1 [∫ t 0 fˆ(τ, λ)e−a 2λ2(t−τ)dτ ] = F−1[ϕˆ(λ)] ∗ F−1[e−a2λ2t] + ∫ t 0 F−1[fˆ(τ, λ)] ∗ F−1[e−a2λ2(t−τ)]dτ = ϕ(x) ∗ 1 2a √ pit e − x 2 4a2t + ∫ t 0 f(τ, x) ∗ 1 2a √ pi(t− τ)e − x 2 4a2(t−τ)dτ = 1 2a √ pit ∫ +∞ −∞ ϕ(ξ)e− (x−ξ)2 4a2t dξ + 1 2a √ pi ∫ t 0 ∫ +∞ −∞ f(τ, ξ)√ t− τ e − (x−ξ)2 4a2(t−2) dξdτ, �m F−1[e−a 2λ2t] = 1 2pi ∫ +∞ −∞ e−a 2λ2teiλxdλ = 1 pi ∫ +∞ 0 e−a 2λ2t cos λxdλ = 1 2a √ pit e − x 2 4a2t , hRH�sS:Rx$-mW%�R��&� � 3.2 ���~' +�4+ ∆2u = 0 (−∞ < x < +∞, y > 0) u(x, 0) = f(x) lim x2+y2→+∞ u(x, y) = 0 Æ 2 x v F �y , X uˆ(λ, y) = F [u(x, y)]. &�/:7��i4+{−λ2uˆ+ ∂2uˆ ∂y2 = 0 uˆ|y=0 = fˆ(λ), uˆ|y=+∞ = 0. 7�.a0 uˆ(λ, y) = Aeλy +Be−λy 80 !��-℄ uˆ|y=0 = A+B = fˆ(λ), uˆ|y=+∞ = 0, & λ > 0 � A = 0; λ < 0 � B = 0, �3 uˆ(λ, y) = fˆ(λ)[h(−λ)eλy + h(λ)e−λy ] = fˆ(λ)e−|λ|y. v6 P& u(x, y) = f(x) ∗ F−1[e−|λ|y] = f(x) ∗ 1 pi y x2 + y2 = 1 pi ∫ +∞ −∞ yf(ξ) (x− ξ)2 + y2dξ, �m F−1[e−|λ|y] = 1 2pi ∫ +∞ −∞ e−|λ|yeiλxdλ = 1 pi ∫ +∞ −∞ e−λy cos λxdλ = 1 pi y x2 + y2 . 2 � 1. F y9ZMi��qeA��|\�fs�uH8��|��9ZH9lH O�7�x (−∞,+∞). 2. ^R9Z?SI4m�9�>��;�pC���L�?Twi��QoE P��oEP�~ÆDj� � 3.1 Æ} ���� � 3.2 Æ} ���� � 3.3 Æ} ���� � 3.3 J0�y}/Lf'�h<�'��$4+ ∂u ∂t = a2 ∂2u ∂x2 , (x > 0, t > 0) ∂u ∂x ∣∣∣∣ x=0 = q u|t=0 = 0 81 Æ 0*0) II k�-℄�7�mDU x '�$04_�2 x v +"y uˆ(t, λ) = Fc[u(t, x)] = ∫ +∞ 0 u(t, x) cos λxdλ. � u|x=+∞ = 0, ∂u∂x ∣∣∣ x=+∞ = 0, ^T Fc [ ∂u ∂t ] = ∂uˆ ∂t , Fc [ ∂2u ∂x2 ] = ∫ +∞ 0 ∂2u ∂x2 cos λxdx = [ ∂u ∂x cos λx+ λu sinλx ]+∞ 0 − λ2 ∫ +∞ 0 u cos λxdx = −q − λ2uˆ. C&�I$|t'-a4+ { ∂uˆ ∂t + a2λ2 = −a2q uˆ|t=0 = 0 a& uˆ(t, λ) = q λ2 (e−a 2λ2t − 1). S +"y '6GA� u(t, x) = 2 pi ∫ +∞ 0 uˆ(t, λ) cos λxdλ = 2 pi ∫ +∞ 0 q λ2 (e−a 2λ2t − 1) cos λxdλ, T ∂u ∂t = 2 pi ∫ +∞ 0 −a2qe−a2λ2t cos λxdλ = − aq√ pit e − x 2 4a2t �3 u(t, x) = u(t, x)− u(0, x) = − aq√ pi ∫ t 0 1√ τ e − x 2 4a2τ dτ v s!P y = x 2a √ τ , & u(t, x) = −qx√ pi ∫ +∞ x 2a √ t 1 y2 e−y 2 dy. 82 2 � 1.5��uH�| ���Yx��9Z�tg�KJH8f}b_P��U � x HFzdz� 2. ����9ZH�RTwik�R<^RB� �G��KJH8f}b (oE�QoEhi) _ +∞ Jr� 0 Hav� 3. qN9Zi�X? F yN9ZW=:�\� `{^R� � 3.4 Æ} ���� � 3.4 �h<�'��$4+ ∂u ∂t = a2 ∂2u ∂x2 , (x > 0, t > 0) u|x=0 = f(t), lim x→+∞ u = 0 u|t=0 = 0 Æ 2�QL4+�h2 xve: P�Ih2 tv L �y . bn�RMa� w u¯(p, x) = L[u(t, x)], X4+h 0>I$'�i4+{ pu¯ = a2 ∂2u¯ ∂x2 u¯|x=0 = f¯(p), u¯|x=+∞ = 0. 7�.a0 u¯(p, x) = Ae √ p a x +Be− √ p a x, !��b-℄ u¯|x=+∞ = 0, T A = 0; u¯|x=0 = B = f¯(p), T u¯(p, x) = f¯(p)e− √ p a x. ��h& u(t, x) = L−1[f¯(p)e− √ p a x] = L−1[f¯(p)] ∗ L−1[e− √ p a x] = f(t) ∗ x 2a √ pit3 e − x 2 4a2t = x 2a √ pi ∫ t 0 f(t− z)√ τ3 e − x 2 4a2τ dτ. 83 oRr8 3.4 '`F�S,YXm h &QL4+ ∂u ∂t = a2 ∂2u ∂x2 (t > 0, x > 0) u|x=0 = ϕ(t) u|t=0 = ψ(x) 'a� � 3.5 T;ÆZ�: .7�'QL4+ ∂2u ∂t2 = a2 ∂2u ∂x2 , (t > 0, 0 < x < l) u|x=0 = 0, ∂u ∂x ∣∣∣∣ x=l = A sinωt u|t=0 = 0, ∂u ∂t ∣∣∣∣ t=0 = 0 Æ b�9���b-℄'QL4+��)a^�b-℄��OMhR:l s5ag�bn�R L �y a� w u¯(p, x) = L[u(t, x)], T p2u¯ = a2 ∂2u¯ ∂x2 (0 < x < l) u¯|x=0 = 0, ∂u¯ ∂x ∣∣∣∣ x=p = Aω p2 + ω2 . ag�h& u¯ = Aaω sh x a p p(p2 + ω2)ch l a p . S L ! P'6GA�Jx$mz u(t, x) = 1 2pii ∫ σ+i∞ σ−i∞ u¯(p, x)eptdp = ∑ k Res[u¯(p, x)ept, pk], �m {pk} 0 u¯ept \ p �~�'&TBp�*� bn u¯eptTh��* p0 = 0,JVT* p˜±1 = ±iω,JVT* p±k = ±i(2k − 1)api2l , k = 1, 2, · · ·. Res[u¯ept, p0] = 0, 84 Res[u¯ept, p˜1] + Res[u¯e pt, p˜−1] = Aaωsh (x a − p ) ePt p(p + iω)ch ( l a p ) ∣∣∣∣∣∣∣∣ p=iω + Aaωsh (x a p ) ePt p(p− iω)ch ( l a p ) ∣∣∣∣∣∣∣∣ p=−iω = 2Re  Aaωsh (x a p ) ePt p(p+ iω)ch ( l a p ) ∣∣∣∣∣∣∣∣ p=iω  = Aa ω cos ωl a sin ωx a sinωt Res[u¯ept, pk] + Res[u¯e pt, p−k] = 2Re  Aaωsh (x a p ) ePt p(p2 + ω2) · 1( ch l a p )′ ∣∣∣∣∣∣∣∣∣ p=i (2k−1)api 2l  = (−1)k−116Aaωl2 (2k − 1)pi[4l2ω2 − a2(k − 1)2pi2] sin (2k − 1)pix 2l sin (2k − 1)piat 2l . � 3.6 5��:'�� . ∂2u ∂t2 = a2 ∂2u ∂x2 + f(t, x), (t > 0,−∞ < x < +∞) u|t=0 = 0, ∂u ∂t ∣∣∣∣ t=0 = 0. Æ X u¯(p, x) = L[u(t, x)], 2��i4+v L �y , & p2u¯ = a2 ∂2u¯ ∂x2 + f¯(p, x), (−∞ < x < +∞) Æ8� u|x=±∞ = 0, �3 u¯|x=±∞ = 0. b��p�'�/:7��i4+�hM[ R F �y a� X ̂¯u(p, λ) = F [u¯(p, x)], T!$7� p2̂¯u = −a2λ2̂¯u+ ̂¯f(p, λ) a& ̂¯u(p, λ) = ̂¯f(p, λ) p2 + a2λ2 , 85 6 P�h& u = L−1[F−1[̂¯u(p, λ)]] = F−1 [ L−1 [ ̂¯f(p, λ) p2 + a2λ2 ]] = F−1 [ L−1 [ 1 p2 + a2λ2 ] ∗ L−1[̂¯f(p, λ)]] = F−1 [ sin aλt aλ ∗ f̂(t, λ) ] = F−1 [∫ t 0 f̂(t− τ, λ)e iaλτ − e−iaλτ 2iaλ dτ ] = 1 2a ∫ t 0 { F−1 [ f̂(t− τ, λ) iλ eiaλτ ] − F−1 [ f̂(t− τ, λ) iλ e−iaλτ ]} dτ = 1 2a ∫ t 0 [∫ x+aτ 0 f(t− τ, ξ)dξ − ∫ x−aτ 0 f(t− τ, ξ)dξ ] dτ = 1 2a ∫ t 0 ∫ x+aτ x−aτ f(t− τ, ξ)dξdτ, bnR%tA� F−1 [ f̂(λ) iλ ] = ∫ x 0 f(ξ)dξ, F−1[f̂(λ)eiλx0 ] = f(x+ x0). 2 � L y9Zi��qeA��|�[V�|�n℄6;�� F y9Z ~� � 3.7 �1iZm'��$4+{ ∂u ∂t = a2∆3u = a 2 ( ∂2u ∂x2 + ∂2u ∂y2 + ∂2u ∂z2 ) , (t > 0,−∞ < x, y, z < +∞) u|t=0 = ϕ(x, y, z) Æ b��iZ'4+�h23l sv�1 Fourier y , UK�2 x, y, z v Fourier P� w û(t, λ, µ, ν) = F3[u(t, x, y, z)] = ∫∫∫ +∞ −∞ u(t, x, y, z)e−i(λx+µy+νz)dλdµdν, 86 ^ F3 [ ∂2u ∂x2 ] = (iλ)2û, F3 [ ∂2u ∂y2 ] = (iµ)2û, F3 [ ∂2u ∂z2 ] = (iν)2û. & û |t'-a4+ { ∂û ∂t = −a2(λ2 + µ2 + ν2)û û|t=0 = ϕ̂(λ, µ, ν), a& û(t, λ, µ, ν) = ϕ̂(λ, µ, ν)e−a 2(λ2+µ2+ν2)t, K�2 λ, µ, ν v F !6 P�?efSA�U& u(t, x, y, z) = 1 (2pi)3 ∫∫∫ +∞ −∞ ϕ̂(λ, µ, ν)e−a 2(λ2+µ2+ν2)tei(λx+µy+νz)dλdµdν = ϕ(x, y, z) ∗ F−1[e−a2λ2t]F−1[e−a2µ2t]F−1[e−a2ν2t] = ϕ(x, y, z) ∗ 1 (2a √ pit)3 e −x 2+y2+z2 4a2t = 1 (2a √ pit)3 ∫∫∫ +∞ −∞ ϕ(ξ, η, ζ)e− (x−ξ)2+(y−η)2+(z−ζ)2 4a2t dξdηdζ. � 3.1 a�i4+{ ∂u ∂t + a ∂u ∂x = f(t, x) (t > 0,−∞ < x < +∞) u|t=0 = ϕ(x) Æ} v F �y & { ∂û ∂t + iλaû = f̂(t, λ) û|t=0 = ϕ̂(λ), a& û(t, λ) = ϕ̂(λ)e−iλat + ∫ t 0 f̂(τ, λ)e−iλa(t−τ)dτ, F !6 P& u(t, x) = ϕ(x− at) + ∫ t 0 f(τ, x− a(t− τ))dτ, bn F−1[ϕ̂(λ)e−iλat] = 1 2pi +∞∫ −∞ ϕ̂(λ)eiλ(x−at)dλ = ϕ(x− at). ���� 1. �F7�')a�N ��� f(t, x) = et, N\T)a v(t, x) = et, �;T� '�["'75 a� 87 2. �;hR L ! P a� � 3.2 a�i4+ { ∂u ∂t = a2 ∂2u ∂x2 + b ∂u ∂x + cu u|t=0 = ϕ(x) Æ} v F �y & { ∂û ∂t = (−a2λ2 + ibλ+ c)û û|t=0 = ϕ̂(λ) a& û(t, λ) = ϕ̂(λ)e−(a 2λ2−ibλ−c)t, �3 u(t, λ) = ϕ(x) ∗ F−1[e−(a2λ2−ibλ−c)t] = ϕ(x) ∗ 1 2pi e−ct ∫ −−∞+∞e−a2λ2−ibλ+iλxdλ = e−ct 2a √ pit ∫ +∞ −∞ ϕ(x− ξ)e− (ξ−b)2 4a2t dξ ���� �;.G2fI$'!P�A�7�m'J_$=�[v F ! P� � 3.3 a�i4+ ∂2u ∂t2 = a2 ∂2u ∂x2 , (t > 0,−∞ < x < +∞) u|t=0 = ϕ(x), ∂u ∂t ∣∣∣∣ t=0 = ψ(x). Æ} v F �y &  ∂2û ∂t2 = −a2λ2û û|t=0 = ϕ̂(λ), ∂û ∂t ∣∣∣∣ t=0 = ψ̂(λ) =⇒ û(t, λ) = Aeiaλt +Be−iaλt, �m A = 1 2 [ ϕ̂(λ) + ψ̂(λ) iaλ ] , B = 1 2 [ ϕ̂(λ)− ψ̂(λ) iaλ ] . 88 =⇒ u(t, x) = 1 2pi ∫ +∞ −∞ [Aeiaλt +Be−iaλt]eiλxdλ = 1 2 [ 1 2pi ∫ +∞ −∞ ϕ̂(λ)eiλ(x+at)dλ+ 1 2pi ∫ +∞ −∞ ϕ̂(λ)eiλ(x−at)dλ ] + 1 2a [ 1 2pi ∫ +∞ −∞ ψ̂(λ) iλ eiλ(x+at)dλ− 1 2pi ∫ +∞ −∞ ψ̂(λ) iλ eiλ(x−at)dλ ] = 1 2 [ϕ(x+ at) + ϕ(x− at)] + 1 2a ∫ x+at x−at ψ(ξ)dξ, �m�uMJ�js 1 2pi ∫ +∞ −∞ ψ̂(λ) iλ eiλ(x+at)dλ− 1 2pi ∫ +∞ −∞ ψ̂(λ) iλ eiλ(x−at)dλ = 1 2pi ∫ +∞ −∞ ∫ x+at x−at ψ̂(λ)eiλξdξdλ = ∫ x+at x−at 1 2pi ∫ +∞ −∞ ψ̂(λ)eiλξdλdξ = ∫ x+at x−at ψ(ξ)dξ. ���� �F7�m_,9��= f(t, x),�;RS: P a� (S,YXm�kD a ϕ(x) = ψ(x) = 0 ' C�) �\ � 3.6. � 3.4 a�h<�'4+ ∂u ∂t = a2 ∂2u ∂x2 , (t > 0, x > 0) u|x=0 = 0 u|t=0 = ϕ(x). Æ} w û(t, λ) = Fs[u(t, x)] = ∫ +∞ 0 u(t, x) sin λxdx, T  ∂û ∂t = a2 ∫ +∞ 0 ∂2u ∂x2 sinλxdx = a2 [ ∂u ∂x sinλx− λu cos λx ]+∞ 0 − a2λ2 ∫ +∞ 0 u sinλxdx = −a2λ2û û|t=0 = ϕ̂(λ), & û = ϕ̂e−a 2λ2t, 89 �3 u = 2 pi ∫ +∞ 0 ϕ̂(λ)e−a 2λ2t sinλxdλ = 2 pi ∫ +∞ 0 ∫ +∞ 0 ϕ(ξ) sin λξdξe−a 2λ2t sinλxdλ = 1 pi ∫ +∞ 0 ϕ(ξ) ∫ +∞ 0 e−a 2λ2t[cos λ(x− ξ)− cos λ(x+ ξ)]dλdξ = 1 2a √ pit ∫ +∞ 0 ϕ(ξ)[e− (x−ξ)2 4a2t − e− (x+ξ) 2 4a2t ]dξ. ���� 1. �F0*M f(t) Z.��Re: P a{� 2. �F7�m_Y ∂u ∂x =��Re: P a{� � 3.5 R Laplace P a-a4+{ ∂2u ∂x∂y = u− e−(x+y), (x > 0, y > 0) u|x=0 = 0, u|y=0 = xe−x. Æ} 2 x v L ! P�X u¯(p, y) = L[u(x, y)]. S L [ ∂2u ∂x∂y ] = p ∂u¯ ∂y − ∂u ∂y ∣∣∣∣ x=0 = p ∂u¯ ∂y − ∂ ∂y u(0, y) = p ∂u¯ ∂y , T  p ∂u¯ ∂y = u¯− e −y p+ 1 u¯|y = 1 (p+ 1)2 a& u¯(p, y) = e−y (p+ 1)2 , �3 u(x, y) = xe−(x+y). ���� 1. "2 y v L ! P� 2. �;2 x R y ve: Pja+� � 3.6 R Laplace P aQL4+ ∂u ∂t = a2 ∂2u ∂x2 − hu, (x > 0, t > 0), a, h 0e�$ u|x=0 = 0, ∂u ∂x ∣∣∣∣ x=+∞ = 0 u|t=0 = b. 90 Æ} w u¯(p, x) = L[u(t, x)], T ∂2u˜ ∂x2 − p+ h a2 u˜ = − b a2 u˜|x=0 = 0, ∂u˜ ∂x ∣∣∣∣ x=+∞ = 0, a& u˜(p, x) = b p+ h (1− e− q p+h a2 x ), u(t, x) = be−ht − be−htercf ( x 2a √ t ) , �m ercf(x) = 2√ pi ∫ +∞ x e−ξ 2 dξ 0V6�I$. ���� h;R�'75 a� � 3.7 R�1 F ! P a�37� ∆3u = −f(x, y, z), 5�Y�0u+ 3� Æ} w û(λ, µ, ν) = F3[u(x, y, z)], & − (λ2 + µ2 + ν2)û = −f̂(λ, µ, ν), û = f̂ ρ2 , ρ2 = λ2 + µ2 + ν2, u(x, y, z) = F−13 [ f̂ ρ2 ] = f(x, y, z) ∗ F−13 [ 1 ρ2 ] , �m F−13 [ 1 ρ2 ] = 1 (2pi)3 ∫∫∫ +∞ −∞ 1 ρ2 ei(λx+µy+νz)dλdµdν v�w� P�� ν p�\?d r = (x, y, z) 7?�^ λ = ρ sin θ cosϕ, µ = ρ sin θ sinϕ, ν = ρ cos θ, λx+ µy + νz = ρ · r = ρr cos θ, r2 = x2 + y2 + z2. T F−13 [ 1 ρ2 ] = 1 (2pi)3 ∫ +∞ 0 ∫ pi 0 ∫ 2pi 0 eiρr cos θ sin θdϕdθdρ = 1 (2pi)2 ∫ +∞ 0 eiρr cos θ iρr ∣∣∣∣pi 0 dρ = 1 2pi2r ∫ +∞ 0 sin ρr ρ dρ = 1 4pir . 91 �3 u(x, y, z) = 1 4pi ∫∫∫ +∞ −∞ f(ξ, η, ζ)√ (x− ξ)2 + (y − η)2 + (z − ζ)2 dξdηdζ. 92