Complex Numbers Cheat Sheet

COMPLEX NUMBERS A complex number Z may be defined as an ordered pair (x,y), where x and y are real numbers…….. Z = x + iy x , y  R (Real set) . x is the real part and y is the imaginary part of the complex number. Z  x  iy is called the complex conjugate of Z Polar form of the complex number: Substituting x  r cos  and y  r sin 

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Z  r (cos   i sin  )  rei 1  y  Thus Z  rei is the polar form of the complex number where r  x 2  y 2 and   tan    x r is called modulus and θ is called amplitude of the complex. A particular value of θ which satisfies the equation  y tan     and which lies between –π and π is called the principle value of the amplitude.  x Properties of Moduli and Amplitude of complex numbers: |Z1 ± Z2| ≤ |Z1| + |Z2| |Z1 ± Z2| ≥ |Z1| – |Z2| |Z1∙Z2∙Z3……….. Zn| = |Z1|∙|Z2|∙|Z3|………..|Zn| Z Z1  1 Z2 Z2 Z1 = r1eiθ, Z2 = r2eiα………Then |Z1∙Z2| = r1∙r2∙ei(θ + α) Z1 r1 i (  )  e Z 2 r2

  Complex no. purely imaginary 2 If amplitude is 0 or   Complex no. purely real If amplitude is

Analytical functions:  A complex valued function f(Z) is said to be analytic at a point Z0 if f(Z) is differential not only at Z0 but at every point of some neighborhood of Z0. '  A complex valued function f(Z) is said to be analytic in a region R, if f ( Z ) exists at every point of the region R.

NOTE: An analytic function is also known as regular, holomorphic, monogenic function.

Entire Function: A function f(Z) which is analytic everywhere in the complex plane (argand plane) is called an entire function. Cauchy–Riemann (C–R) Equations: The cauchy–Riemann equations are applied to determine whether the given complex valued function f(Z) = u + iv is analytic or not. 1. C.R. equation are given by

du dv du dv   and (or) ux = vy and uy = – vx dx dy dy dx

2. If f(Z) is defined in polar form then the C.R. equations in polar form are given by dv 1 du  dr r d

(or)

du 1 dv  and dr r d

vθ = r∙ur and uθ = –r∙vr

3. Harmonic and conjugate harmonic functions: Any function ф(x,y) satisfying the Laplace equation d 2 d 2   0 is called a harmonic function.  2  0 i.e. dx 2 dy 2 Theorem: If f(Z) = u + iv is an analytic function. Then the real and imaginary parts u and v satisfy Laplace equation. (i.e. u and v are harmonic). Thus, NOTE: The

d 2 u d 2u d 2v d 2v   0  0 and dx 2 dy 2 dx 2 dy 2 polar form of laplace equations are given by r2urr + rur + uθθ = 0 and r2vrr + rvr + vθθ = 0.

Properties of Analytic Function: 1. If f(Z) and g(Z) are analytic, then f ( Z )  g ( Z ) , f ( Z ) g ( Z ) and

f (Z ) are also analytic provided g(Z) ≠ 0. g (Z )

2. If f(Z) is analytic, then it is continuous i.e., (Analyticity  Differentiability  Continuity) 3. The derivative of an analytic function is also analytic. 4. If f(Z) = u + iv is analytic, then the family of curves u(x,y) = C1 and v(x,y) = C2 form an orthogonal system i.e., u(x,y) = C1 are orthogonal trajectories of v(x,y) = C2 and vice versa. 5. The real part u(x,y) of an analytic function f(Z) = u(x,y) + iv(x,y) is known as the conjugate harmonic function and is uniquely determined upto an arbitrary real additive constant. NOTE: The harmonic conjugate here is not to be confused with the conjugate Z  x  iy .