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Chapter 2: Complex Numbers I. Background Complex number (z):

Ch. 2- Complex Numbers>Background. Chapter 2: Complex Numbers I. Background Complex number (z): z = A + iB where (note: engineers use ) “real part”: Re{z} = A “imaginary part”: Im{z} = B (both A and B are real numbers) Graphical Representation:

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Chapter 2: Complex Numbers I. Background Complex number (z):

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  1. Ch. 2- Complex Numbers>Background • Chapter 2: Complex Numbers • I. Background • Complex number (z): • z = A + iB where (note: engineers use ) • “real part”: Re{z} = A • “imaginary part”: Im{z} = B • (both A and B are real numbers) • Graphical Representation: • ex: 2+3i • 3-4i

  2. Ch. 2- Complex Numbers>Background • Polar vs. rectangular coordinates: • Rectangular coordinates: z = A + iB • Polar coordinates: z = reiθ • (note: |z| ≡ r , z = θ) • To move between the two: • A + iB = (r cosθ) + i (r sinθ) = reiθ • because eiθ= cosθ + sinθ (see section 7-9 for proof) • so: • θ is in radians, r ≥ 0 • ex: z=3-i4 (write in polar coordinates)

  3. Ch. 2- Complex Numbers>Background • Beware when using • Your calculator can’t tell the • difference between θ and θ+π. • It should always find . • You may need to add π to get to • the correct quadrant! • ex: z = -3 + i4 • In practice, polar form reiθ is often easier to use because it’s easier to • differentiate and integrate reiθ.

  4. Ch. 2- Complex Numbers>Algebra>Complex Conjugate II. Algebra with complex numbers 1) Complex conjugate: Let z = A + iB Then is the complex conjugate. In polar coordinates: z = reiθ z* = re-iθ Why? Recall: z = reiθ = r (cosθ + isinθ) z* = r (cosθ - isinθ) = r (cos(-θ) + isin(-θ)) =re-iθ ex: 1) z = 2 + i3 2) z = 3 – i4 3) z = 5ei(/3) Note: z z* = |z|2 (Proof in a minute…)

  5. Ch. 2- Complex Numbers>Algebra>Addition For the remaining examples: Let z1 = 2 + i3  r = 3.6, (θ) = 0.98 rad  z1 = 3.6ei(0.98) z2 = 4 – i2  r = 4.5, (θ) = -0.46 rad  z1 = 4.5ei(-0.46) 2) Addition: (A + iB) + (C + iD) = (A+C) + i(B+D) there is no easy way to do this in polar coordinates: ex: z1 + z2 (in rectangular coordinates) ex: z1 + z2 (in polar coordinates)

  6. Ch. 2- Complex Numbers>Algebra>Multiplication 3) Multiplication: (A + iB) (C + iD) = AC + i(AB) + i(BC) – BD (term by term) ex: z1•z2 (rectangular) ex: z1•z2 (polar)

  7. Ch. 2- Complex Numbers>Algebra>Multiplying by Complex Conjugate 4) Multiplying by complex conjugate: ex: z1z1* |z1|2 This is true for any complex number: |z1|2 = z1z1* Proof: Let z = A + iB z z*= (A + iB) (A – iB) = A2 – iAB + iBA + B2 = A2 + B2 = r2 = |z|2

  8. Ch. 2- Complex Numbers>Algebra>Division 5) Division: (more difficult in rectangular form) ex: (polar) ex: (rectangular)

  9. Ch. 2- Complex Numbers>Algebra>More Complex Conjugates & Complex Equations 6) More complex conjugates: Say I want the complex conjugate of a messy equation: Change all i  -i 7) Complex equations: (A + iB) = (C + iD) iff A = C and B = D ex: z1 + z2 = x + i(3x + y)

  10. Ch. 2- Complex Numbers>Algebra>Powers 8) Powers: do these in polar form: rectangular form  switch to polar first ex: z12 (polar) ex: z12 (rectangular)

  11. Ch. 2- Complex Numbers>Algebra>Roots 9) Roots: Polar coordinates: ex: Check by changing back to rectangular coordinates. Find another root (add 2 to ). Convert back to rectangular coordinates:

  12. Ch. 2- Complex Numbers>Algebra>Roots In general: has n possible roots! ex: ex:

  13. Ch. 2- Complex Numbers>Algebra>Complex Exponentials 10) Complex Exponentials: let z = x + iy then ez = ex+iy = exeiy = ex(cosy + isiny) ex:

  14. Ch. 2- Complex Numbers>Algebra>Trig Functions 11) Trig Functions:

  15. Ch. 2- Complex Numbers>Algebra>Trig Functions This is very useful for derivatives and integrals: ex: ex: Good Trick:

  16. Ch. 2- Complex Numbers>Algebra>Hyperbolic Functions 12) Hyperbolic Functions: ↔ Usual sin/cos functions Hyperbolic functions (entirely real) Likewise: etc…

  17. Ch. 2- Complex Numbers>Algebra>Natural Logarithm 13) Natural logarithm: Note: i = i(+2) = i(+4) = … So ln(z) has infinitely many solutions: ln(z) = ln(r) + i = ln(r) + i(+2) = … ‘Principal solution’ has .

  18. Ch. 2- Complex Numbers>Example: RLC Circuit Physics Example: RLC Circuit  applied emf Then Find I0 & Φ and Z (the complex impedance). From Physics 216: What is the impedance? (yuck!)

  19. Ch. 2- Complex Numbers>Example: RLC Circuit It’s easier to do this:

  20. Ch. 2- Complex Numbers>Example: RLC Circuit What’s the physically real I? where Recall our actual driving voltage: We wrote this as . Only has any physical reality. So, same goes for current: The physically real current is with same Io & Φ as above.

  21. Ch. 2- Complex Numbers>Example: RLC Circuit Resonance Defn: The frequency at which Z is entirely real: So, at resonance ωR: (resonance frequency) Note that for this circuit, I0 is max when ω=ωR: (at resonance) See plot of I0 vs. ω.

  22. Ch. 2- Complex Numbers>Example: RLC Circuit Complex Impedances We found: “reactance” or where “complex impedance” Complex impedances behave just like resistors in series and parallel.

  23. Ch. 2- Complex Numbers>Example: RLC Circuit From Physics 216: Instead, it’s easier to use complex notation:

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