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3.1 The need for complex numbers

Chapter 3 Complex numbers and hyperbolic functions. 3.1 The need for complex numbers. no real solutions. define complex number , and. the real part x is denoted by Re(z) the imaginary part y is denoted by Im(z). can be written as. 3.2 Manipulation of complex numbers.

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3.1 The need for complex numbers

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  1. Chapter 3 Complex numbers and hyperbolic functions 3.1 The need for complex numbers no real solutions define complex number , and the real part x is denoted by Re(z) the imaginary part y is denoted by Im(z) can be written as 3.2 Manipulation of complex numbers • addition and subtraction

  2. Chapter 3 Complex numbers and hyperbolic functions • modulus and argument the modulus of z the argument of z Ex: • multiplication • special points

  3. Chapter 3 Complex numbers and hyperbolic functions • complex conjugate is a reflection of in the real axis • properties of complex conjugate

  4. Chapter 3 Complex numbers and hyperbolic functions 3.3 Polar representation of complex numbers • complex exponential function for , is real Euler’s equation

  5. Chapter 3 Complex numbers and hyperbolic functions • multiplication and division in polar form and multiplication division 3.4 de Moivre’s theorem

  6. Chapter 3 Complex numbers and hyperbolic functions Ex. 1: Express and in terms of powers of and Ex. 2: • find the nth roots of unity k is an integer

  7. Chapter 3 Complex numbers and hyperbolic functions Ex. : Find the solutions to the equation Imz Rez Ex. : The three roots of are Proof:

  8. Chapter 3 Complex numbers and hyperbolic functions Ex: Solve the polynomial equation (1) (2) (3)

  9. Chapter 3 Complex numbers and hyperbolic functions 3.5 Complex logarithms complex powers • note: multivalued single valued is the principal value of Ex: Ex: is real

  10. Chapter 3 Complex numbers and hyperbolic functions 3.6 Applications to differentiation and integration Ex: Find the derivative with respect to of set Ex: Evaluate the integral

  11. Chapter 3 Complex numbers and hyperbolic functions 3.7 Hyperbolic functions • definition:

  12. Chapter 3 Complex numbers and hyperbolic functions • hyperbolic-trigonometric analogies

  13. Chapter 3 Complex numbers and hyperbolic functions • identities of hyperbolic functions • solving hyperbolic equations Ex: Solve the hyperbolic equation (1) (2)

  14. Chapter 3 Complex numbers and hyperbolic functions • inverse of hyperbolic functions Ex: Find a closed-form expression for the inverse hyperbolic function (1) (2) (3) (1)

  15. Chapter 3 Complex numbers and hyperbolic functions (2) (3)

  16. Chapter 3 Complex numbers and hyperbolic functions • calculus of hyperbolic functions

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