1 / 16

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.

yosef
Download Presentation

3.1 The need for complex numbers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related