Characteristic Equations with Complex Roots

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Complex Numbers

Definition: Complex Number

A complex number is a number that can be displayed in the form a+bi. The real part of complex number is aR and the imaginary part of complex number is bR. The imaginary unit i satisfies i2=1.

Conjugate Pairs of Complex Numbers

Definition: Complex Conjugate Pairs

Let z=a+bi, then the complex conjugate of z is z=z=abi.

Multiplication of Imaginary Units

Multiplication of Complex Numbers

Let z1=a1+b1i and z2=a2+b2i, then the product z1z2 is defined as

z1z2=(a1+b1i)(a2+b2i)xxx=a1a2+a1b2i+a2b1i+(b1i)(b2i)xxx=a1a2+a1b2i+a2b1i+b1b2i2xxx=a1a2b1b2+(a1b2+a2b1)i

Example 05

(32i)(1+4i)=3+(8i2)+(12i2i)=11+10i

Division of Complex Numbers

Let z1=a1+b1i and z2=a2+b2i, then the division z1z2 is defined as z1z2(z2z2)

Example 06

11+2i=11+2i(12i12i)=12i5=152i5

Reciprocals of Imaginary Units

1i=1i(ii)=i1=i

Trigonometry

Euler's Formula

Definition: Euler's Formula

eix=cos(x)+isin(x).

Investigation 02

Prove Euler's Formula using the following Power Series:

  1. ex=n=0xnn!

  2. cos(x)=n=0(1)nx2n(2n)!

  3. sin(x)=n=0(1)nx2n+1(2n+1)!

Investigation 03

Evaluate eiπ.

Here is a poem written about eiπ: http://www.3blue1brown.com/poems/epii.

Here is the poem being read by the author of the poem: https://www.youtube.com/watch?v=zLzLxVeqdQg.

de Moivre's Formula

Definition: de Moivre's Formula

[cos(x)+isin(x)]n=cos(nx)+isin(nx).

Investigation 04

Prove de Moivre's Formula.

Investigation 05

  1. Use de Moivre's formula to find an equivalent expression for: sin(2x).

  2. Use de Moivre's formula to find an equivalent expression for: cos(2x).

  3. Use de Moivre's formula to find an equivalent expression, in terms of cosine only, for: cos(3x).

  4. Use de Moivre's formula to find an equivalent expression, in terms of cosine only, for: cos3(x).

  5. Evaluate: cos3(x)dx.

Investigation 06

  1. Prove: cos(x)=eix+eix2.

  2. Prove: sin(x)=eixeix2i.

  3. Find an equivalent expression for: sin(x+y).

  4. Find an equivalent expression for: cos(x+y).

  5. Find an equivalent expression for: sin(xy).

  6. Evaluate: sin(4x)cos(5x)dx

Investigation 07

  1. Rewrite e(1+2i)x in terms of ex, cos(x), and sin(x).

  2. Use the above to evaluate: excos(2x)dx

Hyperbolic Functions

Investigation 08

  1. Show ex=cosh(x)+sinh(x).

  2. Show ex=cosh(x)sinh(x).

  3. Show ddxcosh(x)=sinh(x).

  4. Show ddxsinh(x)=cosh(x).

  5. Show ddxtanh(x)=sech2(x).

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Last Modified: Monday, 6 September 2020 13:33 EDT