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01 bilangan kompleks

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  • 1. Analisis Kompleks
    0
    Bilangan Kompleks
    AnalisisKompleks
    By: SitiKomsiyah, M.Si
  • 2. 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
  • 3. 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
  • 4. 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
  • 5. 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
  • 6. 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
  • 7. 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
  • 8. Analisis Kompleks
    7
    Im
    y
    z = x + i y
    Re
    x
    BilanganKonjugatKompleks(Bil. KompleksSekawan)
    z = x + i y
    = x - i y  konjugat dari z
  • 9. Analisis Kompleks
    8
    OperasiAritmetikBil. KompleksSekawan
  • 10. Analisis Kompleks
    9
    Contoh
    z1= 2 – i 3  z1= 2 + i 3
    z2= 1 + i 3  z2= 1 – i 3
    Hitunglah :
  • 11. 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 )
  • 12. Analisis Kompleks
    11
    Contoh
  • 13. Analisis Kompleks
    12
    ASSIGNMENT 01
  • 14. Analisis Kompleks
    13
  • 15. Analisis Kompleks
    14
  • 16. You must study hard…
    Analisis Kompleks
    15