Background Complex Analsis (1A)
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Complex Numbers Complex Analsis (BGND.1A) 3
Complex Numbers i = 1 (3, 2) two real numbers 2-d coordinate i 2 i 3 = 1 = i i 2 i = i 3 + i2 one complex number with real part of 3 and imaginar part of 2 i 4 = 1 i 2 i 2 = +1 imaginar (3,2) i i i 3 + i2 1 1 real i i i Complex Analsis (BGND.1A) 4
Coordinate Sstems (x, ) x = r cos = r sin (a) Cartesian Coordinate Sstem θ r (r,θ) r = x 2 + 2 tan θ = x (b) Polar Coordinate Sstem Complex Analsis (BGND.1A) 5
Coordinate Sstems and Complex Numbers (x, ) x + i (a) Cartesian Coordinate Sstem (c) Complex Number (I) imaginar r (r,θ) r e i θ θ θ real (b) Polar Coordinate Sstem (d) Complex Number (II) Complex Analsis (BGND.1A) 6
Complex Numbers imaginar x = r cos r = x 2 + 2 x + i = r sin tan θ = x real x + i = r cos θ + i r sin θ = r(cos θ + i sinθ) = r e iθ (c) Complex Number (I) imaginar r e i θ θ real e iθ = cosθ + isin θ (d) Complex Number (II) Complex Analsis (BGND.1A) 7
Euler's Formula (1) e i π/ 2 e iθ = cosθ + isin θ i i i e i π 1 1 e 0 x R{e iθ } = cos θ I{e i θ } = sin θ i i e i 3π/2 i e 0 = +1 + 0i = +1 = e i 2 π e i π/ 2 = 0 + 1i = +i = e = 1 + 0 i e i π = 0 1i e i 3 π/2 i 3π/ 2 = 1 = e i π i π /2 = i = e Complex Analsis (BGND.1A) 8
Euler's Formula (2) e +iθ = cos θ + isin θ e iθ = cosθ isin θ e i π/2 i3 π/2 e i3 π/ 4 e e i π/ 4 i5 π/ 4 e i7 π/ 4 e e i π e 0 e i π e 0 e i5 π/ 4 i7 π/ 4 e e i3 π/2 e i3 π/ 4 π /4 e e π /2 Complex Analsis (BGND.1A) 9
The Euler constant e d d x ax = lim h 0 a x h a x h = a x lim h 0 a h 1 h a x such a, we call e lim h 0 a h 1 h a h a 0 = 1 lim f ' 0 = 1 h 0 h 0 = 1 d d x ex = e x e = 2.71828 f x = e x f ' x = e x f ' ' x = e x Complex Analsis (BGND.1A) 10
The Euler constant e d d x ex = e x e = 2.71828 f x = e x f ' x = e x f ' ' x = e x lim h 0 a h 1 h f ' 0 = 1 http://en.wikipedia.org/wiki/derivative = 1 iif a = e lim h 0 a h a 0 h 0 = 1 Functions f(x) = a x are shown for several values of a. e is the unique value of a, such that the derivative of f(x) = a x at the point x = 0 is equal to 1. The blue curve illustrates this case, ex. For comparison, functions 2 x (dotted curve) and 4 x (dashed curve) are shown; the are not tangent to the line of slope 1 and -intercept 1 (red). Complex Analsis (BGND.1A) 11
The Derivative of a x a x = e ln ax = e x ln a d d x {ax } = d d x {e x ln a } = {e x ln a } d d x d d x {ax } = {a x } ln a {x ln a} d d x {ex } = {e x } ln e = {e x } Complex Analsis (BGND.1A) 12
Absolute Values and Arguments z = r e i θ = r(cosθ + i sin θ) r sin θ sin θ 1 cos θ r r cos θ absolute value argument, phase z = r arg (z) = θ r e iθ = r e i θ = r cos 2 θ + sin 2 θ arg(r e iθ ) Complex Analsis (BGND.1A) 13
Computing Complex Number Argument x < 0 x > 0 atan ( x ) + π atan ( x ) [0, +π] x > 0 x < 0 atan ( x ) π atan ( x ) [0, π] atan ( atan(/x) ) atan2(, x) x π 2 +π π 2 π π atan ( x ) + π π 2 π π 2 atan ( x ) π Complex Analsis (BGND.1A) 14
sin and cos e i = cos i sin e +i θ = cosθ + i sinθ e i = cos i sin R {e i } = cos = ei e i 2 I{e i } = sin = ei e i 2i e i θ = cosθ + i sinθ e i θ = cos θ i sinθ e +i θ = cosθ + i sinθ Complex Analsis (BGND.1A) 15
Complex Exponential e j t = cos t j sin t x real Top View z phase θ = ωt time 2π time imag Front View z imag x real time imag Side View z time x real Complex Analsis (BGND.1A) 16
Conjugate Complex Exponential I I R R t t e + j ω 0t = cos (ω 0 t) + j sin (ω 0 t) e j ω t = cos (ω 0 t) j sin (ω 0 t) e + j t = cos (t) + j sin (t) (ω 0 = 1) e jt = cos (t) j sin (t) (ω 0 = 1) Complex Analsis (BGND.1A) 17
Cos(ω 0 t) e + jt = cos (t) + j sin (t) (e + jt + e j t ) = 2cos (t) e jt = cos (t) j sin (t) I R t x(t) = A cos (ω 0 t) = A 2 e+ j ω 0t + A 2 e j ω 0t Complex Analsis (BGND.1A) 18
Sin(ω 0 t) e + jt = cos (t) + j sin (t) (e + jt e j t ) = 2 j sin (t) I e j t = cos (t) + j sin (t ) R t x(t) = A sin (ω 0 t) = A 2 j e+ j ω 0 t A 2 j e j ω 0 t Complex Analsis (BGND.1A) 19
Complex Power Series e x = 1 x x2 2! x3 3! x4 4! cos x = 1 x2 2! x 4 4! x6 6! sin x = x x3 3! x5 5! x7 7! e i = 1 i i 2 2! i 3 3! i 4 4! i 5 5! = 1 i 2 2! i 3 3! 4 4! i 5 5! = 1 2 2! 4 4! i 3 3! 5 5! e i = cos i sin Complex Analsis (BGND.1A) 20
Talor Series f x = f a f a x a f a 2! x a 2 f n a n! x a n f a f ' a f ' ' a f 3 a f x x = a Complex Analsis (BGND.1A) 21
Maclaurin Series f x = f a f a x a f a 2! x a 2 f n a n! x a n f x = f 0 f 0 x f 0 2! x 2 f n 0 n! x n f 0 f ' 0 f ' ' 0 f 3 0 x = 0 f x Complex Analsis (BGND.1A) 22
Power Series Expansion f x = f 0 f 0 x f 0 2! x 2 f n 0 n! x n cos x = 1 x2 2! x 4 4! x6 6! sin x = x x3 3! x5 5! x7 7! e x = 1 x x2 2! x3 3! x4 4! Complex Analsis (BGND.1A) 23
Complex Arithmetic z = a + i b w = c + i d z+w = (a+c) + i(b+d) z = a + i b w = c + i d z w = (a c) + i(b d) z = a + i b w = c + i d z w = (ac bd ) + i(ad+bc) z = a + i b w = c + i d z w = ( a + i b c + i d ) = ( a + i b c + i d )( c i d c i d ) = ( ac+bd c 2 +d 2 ) + i ( ad +bc ) c 2 +d 2 Complex Analsis (BGND.1A) 24
Complex Conjugate z = x + i = R{z} + i I{z} z = x i = R{z} i I{z} z + w = z + w z w = z w R{z} = 1 (z + z) 2 z w = z w I{z} = 1 (z z) 2i ( z w ) = z w z z = (x + i )(x i ) = x 2 + 2 1 z = 1 z z z = z z z = z = z x 2 + 2 z 2 Complex Analsis (BGND.1A) 25
Complex Power (1) a = e log e a = e ln a a b = (e log ea ) b = (e ln a ) b = e b ln a a i b = (e log a e ) i b = (e ln a ) ib ib ln a = e = cos(b ln a) + isin(b ln a) a c+ib = a c (e log e a ) ib = a c (e ln a ) i b = a c e i bln a Complex Analsis (BGND.1A) 26
Complex Power (2) a = e ln a a b = e bln a a i b = e i bln a = [cos(b ln a) + isin(b ln a)] a c+ib = a c e i b ln a = a c [cos(b ln a) + isin(b ln a)] Complex Analsis (BGND.1A) 27
References [1] http://en.wikipedia.org/ [2] M.L. Boas, Mathematical Methods in the Phsical Sciences [3] E. Kreszig, Advanced Engineering Mathematics [4] D. G. Zill, W. S. Wright, Advanced Engineering Mathematics [5] www.chem.arizona.edu/~salzmanr/480a