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01 bilangan kompleks
 

01 bilangan kompleks

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    01 bilangan kompleks 01 bilangan kompleks Presentation Transcript

    • Analisis Kompleks
      0
      Bilangan Kompleks
      AnalisisKompleks
      By: SitiKomsiyah, M.Si
    • Analisis Kompleks
      1
      Definisi
      Bilangan kompleks z adalah :
      Suatu pasangan terurut (x,y) dari bilangan nyata (x,y)
      Notasi :
      z = (x,y) = x + i y
      Dimana :
      x = bagian nyata (real part) dari z
      Y = bagian imajiner (imaginary part) dari z
    • Analisis Kompleks
      2
      Im
      P
      y
      z = x + i y
      Re
      x
      Bidang kompleks
      Definisi
      Re z = x
      Im z = y
      Misalnya :
      Re(4,5) = 4
      Im(4,5) = 5
      z = 4 + i 5
    • Analisis Kompleks
      3
      Im
      y
      A
      z = 2 + i 3
      1
      2
      3
      Re
      1
      2
      3
      4
      x
      Contoh
      Gambarkandalambidangkompleks z = 2 + i 3
    • Analisis Kompleks
      4
      OperasiBilanganKompleks
      z1 = x1 + i y1
      z2 = x2 + i y2
      Penjumlahan
      z1 + z2 = (x1+x2) + i (y1+y2)
      Pengurangan
      z1 - z2 = (x1-x2) + i (y1-y2)
      Perkalian
      z1 . z2 = (x1 + i y1) (x2 + i y2)
      = (x1 x2 – y1 y2 ) + i (x1 y2 + x2 y1 )
      Pembagian
    • Analisis Kompleks
      5
      ContohSoal
      Diketahui : z1 = x1 + i y1 = 4 + i 5
      z2 = x2 + i y2 = -2 - i 3
      Hitunglah :
      z1 + z2
      z1 - z2
      z1 . z2
      z1 / z2
    • Analisis Kompleks
      6
      Sifat-sifatOperasi
      Hukum komutatif
      z1 + z2 = z2 + z1
      z1 . z2 = z2 . z1
      Hukum asosiatif
      (z1 + z2 ) + z3 = z1 + (z2 + z3 )
      (z1 . z2 ) . z3 = z1 . (z2 . z3 )
      Hukum distributif
      z1.(z2 + z3 ) = z1 z2 + z1 z3
      Identitas
      0 + z = z + 0 = z
      1 . z = z . 1 = z
      Invers
      z + (-z) = (-z) + z = 0
    • Analisis Kompleks
      7
      Im
      y
      z = x + i y
      Re
      x
      BilanganKonjugatKompleks(Bil. KompleksSekawan)
      z = x + i y
      = x - i y  konjugat dari z
    • Analisis Kompleks
      8
      OperasiAritmetikBil. KompleksSekawan
    • Analisis Kompleks
      9
      Contoh
      z1= 2 – i 3  z1= 2 + i 3
      z2= 1 + i 3  z2= 1 – i 3
      Hitunglah :
    • Analisis Kompleks
      10
      Im
      y
      z = x + i y
      r

      Re
      x
      BilanganKompleksdalamKoordinatKutub
      x = r cos
      y = r sin 
      Bentuk :
      z = x + i y
      = r cos + i r sin 
      z = r (cos + isin )
    • Analisis Kompleks
      11
      Contoh
    • Analisis Kompleks
      12
      ASSIGNMENT 01
    • Analisis Kompleks
      13
    • Analisis Kompleks
      14
    • You must study hard…
      Analisis Kompleks
      15