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Fungsi Harmonik

Fungsi Harmonik. Oleh : Kelompok 5 Farid Sugiono 070210191156 Akhmad Mukhlis 070210191154 M. Sidik Yusuf 070210191157 M. Sofyan Hadi 070210191140 Malihur Rohma 070210191143 Martha Citra D. 070210191161. Fungsi Harmonik

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Fungsi Harmonik

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  1. Fungsi Harmonik Oleh :Kelompok 5 Farid Sugiono 070210191156 Akhmad Mukhlis 070210191154 M. Sidik Yusuf 070210191157 M. Sofyan Hadi 070210191140 Malihur Rohma 070210191143 Martha Citra D. 070210191161

  2. FungsiHarmonik f(z) = u(x,y) + iv(x,y) analitikpada D maka u dan v mempunyaiderivatifparsialdisemuaorde yang kontinuepada D. Jadidalam D berlaku C-R , ux = vydanuy = –vx Karenaderifatif-derivatifparsialdari u dan v kontinuedalam D, makaberlakuvxy = vyx. Jikadalamux = vydanuy = –vxdiderivatifkanparsialterhadap x dan y maka(x,y) D berlaku uxx + uyy = 0 vxx = vyy = 0

  3. Jika f analitikpada D maka u dan v pada D memenuhipersamaandifferensial Laplacedalam 2 dimensi. u dan v dimana f(z) = u(x,y) + iv(x,y) analitikpadasuatu domain makaf(z) harmonikpada domain tersebut.

  4. Duafungsi u dan v sedemikiansehingga f(z) = u(x,y) + iv(x,y) analitikdalamsuatu domain dinamakanDuaFungsiyang HarmonikKonjugatdalam domain itu. Suatufungsi 2 peubah (riil) ygmemenuhi pers. Laplace disebutfungsi Harmonic (u,v:harmonic function) u : fungsisekawanharmonis v v : fungsisekawanharmonis u

  5. Contoh 3 Diberikan u(x,y) harmonikpada D dantentukanfungsi v yang harmonikkonjugatdengan u = 4xy3 – 4x3y, (x,y) ℂ Jawab : Misaldiklaimkonjugatnyaadalah v(x,y) jadi f(z) = u(x,y) + iv(x,y) analitikpadaℂsedemikiansehinggaberlaku C-R ux = vydanuy = -vx ux = 4y3 – 12x2yvy = 4y3 – 12x2y uy= 12xy2 – 4x3v= y4 – 6x2y2 + g(x) karenavx= –uymaka –12xy2 + g’(x) = –12xy2 + 4x3sehingga g’(x) = 4x3diperoleh g(x) = x4 + C Jadi v = y4 – 6x2y2 + x4 + C

  6. Cara Milne Thomson Cara yang lebihpraktismenentukanfungsiharmonikkonjugatataudarifungsiharmonik u diberikan u(x,y) harmonikpada D andaikan v(x,y) sehingga f(z) = u(x,y)+ iv(x,y) analitikpada D f”(z) = ux(x,y) + ivx(x,y) sesuaipersamaan C-R : f”(z) = ux(x,y) – iuy(x,y) z = x + iydan = x – iysehinggadiperoleh f(z) = ux –iuy

  7. Suatuidentitasdalam z dan , jikadiambil = z maka f’(z) = ux(z,0) – iuy(z,0) Jadi f(z) adalahfungsi yang derivatifnyaux(z,0) – iuy(z,0) kemudiandidapat v(x,y)

  8. Contoh 5 Dari Contoh 3 dengan u= 4xy3 – 4x3y, (x,y) ℂ, jikadiselesaikandenganmenggunakancara Milne Thomson. Jawab : ux = 4y3 – 12x2y uy= 12xy2 – 4x3 f’(z) = ux(z,0) – iuy(z,0) = –i(– 4z3) = 4iz3 sehingga f(z) = iz4 + C f(z) = i(x + iy)4 + C = 4xy3 – 4x3y + i(x4 – 6x2y2 + y4) + C

  9. Thankz

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