MATH1012 - Lecture 26

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MATH1012 - Lecture 25
Skeleton Slides

Even Functions

• A function $f(t)$ is even if $f(-t)=f(t)$
  
• Its graph has vertical reflection symmetry
  • i.e. reflection over the $y$-axis

Odd Functions

• A function $f(t)$ is odd if $f(-t)=-f(t)$
  
• The graph has $180°$ rotational symmetry

Properties of Even and Odd Functions

• The sum of two even functions is even
  • $even+even=even$
• The sum of two odd functions is odd
  • $odd+odd=odd$
• The product of two even functions is even
  • $even\cdot even=even$
• The product of two odd functions is even
  • $odd\cdot odd=even$
• The product of an even and an odd function is odd
  • $even\cdot odd=odd$
• Evenness/oddness is referred to as parity
  • Multiplying by a constant preserves parity
    • $f(t)=even\to c\cdot f(t)=even$
  • Differentiating switches parity
    • $\frac{d}{dx}(even)=(odd)$

Fourier Cosine Series for Even Functions

• If $f(t)$ is an even function with period $2L$, then:
  $a_0=\frac{1}{L}{\int }_{-L}^L\underset{\text{even function}}{\underbrace{f(t)}}dt=\frac{2}{L}{\int }_0^{L}f(t)dt$
  $a_n=\frac{1}{L}{\int }_{-L}^L\underset{\text{even function}}{\underbrace{f(t)\cos \left(\frac{n\pi t}{L}\right)}}dt=\frac{2}{L}{\int }_0^{L}f(t)\cos (\frac{n\pi t}{L})dt$

\underbrace{f(t) \sin\left(\frac{n\pi t}{L}\right)}_{\text{{\emph{odd function}}}} \, dt = 0$$ ### Fourier Sine Series for Odd Functions - If $f (t)$ is an <font color="#00ff52">odd function</font> with period $2L$, then: $$a_0 = \frac{1}{L} \int_{-L}^{L} \underbrace{f(t)}_{\text{{\emph{odd function}}}} \, dt = 0$$ $$a_n = \frac{1}{L} \int_{-L}^{L} \underbrace{f(t) \cos\left(\frac{n\pi t}{L}\right)}_{\text{{\emph{odd function}}}} \, dt = 0$$ $$b_n = \frac{1}{L} \int_{-L}^{L} \underbrace{f(t) \sin\left(\frac{n\pi t}{L}\right)}_{\text{{\emph{even function}}}} \, dt = \frac{2}{L} \int_0^L f(t) \sin(\frac{n\pi t}{L})\, dt$$ ### Half-Range Expansions - Let $f(t)$ be defined on $[0, L]$ - An expansion of $f(t)$ to $[-L, L]$ is called a *half-range expansion* - We can observe two separate forms for odd and even functions: - <font color="#f38ba8">Even half-range expansion:</font> $g(t) = \begin{cases} f(t), & \text{if } 0 \leq t \leq L \\ f(-t), & \text{if } -L \leq t \leq 0 \end{cases}$ - <font color="#a6e3a1">Odd half-range expansion:</font> $h(t) = \begin{cases} f(t), & \text{if } 0 < t < L \\ 0, & \text{if } t = 0 \\ -f(-t), & \text{if } -L < t < 0 \end{cases}$ - Even functions are always continuous! --- > [[MATH1012 - Week 9 (Annotated).pdf|Annotated Slides]]