Variabel Kompleks e Ekstensi_2 Genap 2014 15
-
Upload
muhammad-fachrurozi -
Category
Documents
-
view
221 -
download
4
description
Transcript of Variabel Kompleks e Ekstensi_2 Genap 2014 15
![Page 1: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/1.jpg)
Ch17_1
Contents
• Bilangan Kompleks• Pangkat dan Akar• Himpunan pada Bidang Kompleks • Fungsi Variabel Kompleks• Persamaan Cauchy-Riemann • Fungsi eksponensial dan Logaritma• Fungsi Trigonometri dan hiperbolik • Fungsi Inverse Trigonometri dan hiperbolik
![Page 2: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/2.jpg)
Ch17_2
17.1 Complex Numbers
z = x + iy, the real number x is called the real part and y is called the imaginary part:
Re(z) = x, Im(z) = y
A complex number is any number of the z = a + ib where a and b are real numbers and i is the imaginary units.
DEFINITION 17.1 Complex Number
![Page 3: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/3.jpg)
Ch17_3
x + iy = 0 iff x = 0 and y = 0.
Complex number and areequal, , if and
DEFINITION 17.2 Complex Number
111 iyxz 222 iyxz
21 zz )Re()Re( 21 zz )Im()Im( 21 zz
![Page 4: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/4.jpg)
Ch17_4
Arithmetic Operations
22
22
21212
22
2
2121
2
1
2121212121
212121
212121
222111
)()()()()()(
, Suppose
yxyxxyi
yxyyxx
zz
yxxyiyyxxzzyyixxzzyyixxzz
iyxziyxz
![Page 5: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/5.jpg)
Ch17_5
Complex Conjugate
1 2 1 2
1 2 1 2
1 2 1 2
1 1
2 2
Suppose , , andz x iy z x iy
z z z z
z z z z
z z z z
z zz z
![Page 6: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/6.jpg)
Ch17_6
Two important equations
(1)
(2)
(3)
and izzzzzz
2)Im(,
2)Re(
xiyxiyxzz 2)()(
22222))(( yxyixiyxiyxzz
iyiyxiyxzz 2)()(
![Page 7: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/7.jpg)
Ch17_7
Contents
• Bilangan Kompleks• Pangkat dan Akar• Himpunan pada Bidang Kompleks • Fungsi Variabel Kompleks• Persamaan Cauchy-Riemann • Fungsi eksponensial dan Logaritma• Fungsi Trigonometri dan hiperbolik • Fungsi Inverse Trigonometri dan hiperbolik
![Page 8: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/8.jpg)
Ch17_8
Bentuk Geometrik, bilangan kompleks z
Gambar 17.1 disebut bidang kompleks x-y dan bilangan kompleks z dinyatakan sebagai vektor posisi
![Page 9: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/9.jpg)
Ch17_9
Modulus atau absolut nilai z = x + iy, dinyatakan Oleh │z│, adalah bilangan nyata
DEFINITION 17.3Modulus atau Nilai Absolute
zzyxz 22||
![Page 10: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/10.jpg)
Ch17_10
Example 3
Jika z = 2 − 3i, maka
Seperti pada Gambar 17.2, jumlah dari vektor z1 dan z2 adl vektor z1 + z2. maka
(5)Hasil pada (5) juga dikenal sebagai ketidaksamaan segitiga dan menjangkau setiap penjumlahan berhingga :
(6)dengan (5),
13)3(2 22 z
2121 zzzz
(7) )(
2121
221221
zzzzzzzzzz
nn zzzzzz ...... 2121
![Page 11: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/11.jpg)
Ch17_11
Fig 17.2
![Page 12: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/12.jpg)
Ch17_12
17.2 Pangkat dan Akar
Bentuk Polar dari Fig 17.3,
z = r(cos + i sin ) (1)dimana r = |z| adl modulus dr z dan adl argumen dr z, = arg(z). Jika adl didlm interval − < , ini disebut argumen prinsip/utama, ditulis dgn Arg(z).
![Page 13: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/13.jpg)
Ch17_13
Fig 17.3
![Page 14: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/14.jpg)
Susunan bilangan kompleks z = x + iy dibentuk oleh pasangan bilangan real (x, y) yg tdr dr bagian real dan imajiner, contoh
•pasangan bilangan (2, -3) bilangan kompleks z = 2 - 3i.•Bilangan angka 7, i, dan-5i (7, 0), (0, 1), (0, -5),
bilangan kompleks z = x + iy dengan titik (x, y) dalam bidang koordinat , z.
Vektor • komponen bilangan kompleks dapat diartikan sbg komponen dari vektor• bilangan kompleks z = x + iy sbg vektor posisi dua dimensi, yi. vektor yg berawal dari
titik asal dan berakhir di titik terminal (x, y). • panjang jarak vektor, z dari titik asal ke titik terminal (x, y) disebut modulus.
![Page 15: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/15.jpg)
Definisi Modulus
Sifat bilangan kompleks, z = x + iy
1.
2.
3.
![Page 16: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/16.jpg)
Polar Form of Complex Numbers
calculator will give only angles satisfying
We have to choose θ consistent with the quadrant in which z is located
angles in the first and fourth quadrants.
![Page 17: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/17.jpg)
Ch17_17
Example 1
Solution Dari Gambar 17.4 bahwa titik terletak pada kuadran keempat.
form.polar in 31 Express i
35sin
35cos2
35)arg(,
13tan
23131
iz
z
izr
![Page 18: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/18.jpg)
Ch17_18
Example 1 (2)
In addition, choose that − < , thus = −/3.
)
3sin()
3cos(2 iz
![Page 19: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/19.jpg)
Ch17_19
Fig 17.4
![Page 20: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/20.jpg)
Bentuk polar dari Bilangan kompleks
![Page 21: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/21.jpg)
Argumen z, sudut
argument θ bil. Kompleks yg terletak di dlm interval −π < θ ≤ π disebut nilai prinsip (principal value ) dari arg(z) atau argumen prinsip dari z, ditulis
For example, if z = i, in Figure 1.9 that some values of arg(i) are π/2, 5π/2, −3π/2, and so on, but Arg(i) = π/2
Argumen dari terletak di dlm interval (−π, π), argumen prinsip z adalah Arg(z) = π/6 − π = −5π/6. dengan menggunakan Arg(z) diperoleh bentuk polar bil. kompleks
![Page 22: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/22.jpg)
![Page 23: Variabel Kompleks e Ekstensi_2 Genap 2014 15](https://reader035.fdocuments.net/reader035/viewer/2022062410/563db97e550346aa9a9ddb91/html5/thumbnails/23.jpg)
Ch17_23