Prove derivative of an odd function is even and vise versa.

  • Thread starter yungman
  • Start date
  • #1
5,598
213

Homework Statement


Prove if f(x) is defined on -L< x < L, and if f(x) is odd function on (-L,L), the f'(x) is even function. and vise versa.


Homework Equations



Using Fourier series expansion, f(x) is odd function on (-L,L) can be represented by Fourier sine series expansion. If f(x) is even function on (-L,L), the it can be represented by cosine series.



The Attempt at a Solution



By series expansion of odd and even function, if f(x) is odd and represented by sine series, then the derivative of the sine series is cosine series and it become an even function. And the reverse is true from even to odd by derivative.


Anyone have a better way of proving this than what I have instead of refering to series expansion?
 

Answers and Replies

  • #2
gabbagabbahey
Homework Helper
Gold Member
5,002
7
You could use the limit definition of f'(-x) to show that it is equal to f'(x)
 
  • #3
430
3
Let [itex]n(x) = -x[/itex] so basically just a "negation function". To say that f is even is to say f(x)=f(-x) or in other words [itex]f = f \circ n[/itex], and to say it's odd is to say -f(x)=f(-x) or in other words [itex]-f = f\circ n[/itex].

Now use:
[tex](f \circ n)(x)' = f'(n(x)) n'(x) = -f'(-x)[/tex]
in both cases.
 
  • #4
5,598
213
Let [itex]n(x) = -x[/itex] so basically just a "negation function". To say that f is even is to say f(x)=f(-x) or in other words [itex]f = f \circ n[/itex], and to say it's odd is to say -f(x)=f(-x) or in other words [itex]-f = f\circ n [/itex].

Now use:
[tex](f \circ n)(x)' = f'(n(x)) n'(x) = -f'(-x)[/tex]
in both cases.

Is [tex]f\circ n = f(n(x))=f(-x)?[/tex]
 
Last edited:
  • #5
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
2021 Award
5,042
1,037
Yes it is
 
  • #6
5,598
213
Let [itex]n(x) = -x[/itex] so basically just a "negation function". To say that f is even is to say f(x)=f(-x) or in other words [itex]f = f \circ n[/itex], and to say it's odd is to say -f(x)=f(-x) or in other words [itex]-f = f\circ n[/itex].

Now use:
[tex](f \circ n)(x)' = f'(n(x)) n'(x) = -f'(-x)[/tex]
in both cases.

Let me clarify:

A) Let f(x) be even:

[tex]\Rightarrow f(-x)=f(x)[/tex]

[tex]f(-x)= f(n(x)) =f(x)\Rightarrow [f(n(x))]'=f'(x)[/tex] (1)

[tex] [df(n(x))]' =f'(nx)n'(x)=-f'(n(x))=-f'(-x)[/tex] (2)

(1) and (2)[tex]\Rightarrow f'(x)=-f'(-x)[/tex]

Therefore if f(x) is even, f'(x) is odd.




B) Let f(x) be odd:

[tex]\Rightarrow f(-x)=-f(x)[/tex]

[tex]f(-x)= f(n(x)) =-f(x)\Rightarrow [f(n(x))]'=-f'(x)[/tex] (3)

[tex] [df(n(x))]' =f'(nx)n'(x)=-f'(n(x))=-f'(-x)[/tex] (4)

(3) and (4)[tex]\Rightarrow -f'(x)=-f'(-x)[/tex]

Therefore if f(x) is odd, f'(x) is even



Tell me whether I got this right? Does this apply to integration that integrate of an odd function is an even function plus a constant and vise versa?

Thanks

Alan
 
  • #7
430
3
Tell me whether I got this right?
Yes you did.

Does this apply to integration that integrate of an odd function is an even function plus a constant and vise versa?

Given a function f(x) that is integrable on the appropriate intervals define:
[tex]F(x) = \int_0^x f(t) dt[/tex]
Then you wish to show that f(x) is odd imply F(x) even, and f(x) even imply F(x) odd. In general we have the identity:
[tex]\int_a^b f(t) dt = -\int_{-a}^{-b}f(-t)dt[/tex]
You can use that to write:
[tex]F(-x) = \int_0^{-x} f(t)dt = -\int_0^x f(-t) dt[/tex]
Now you can use f(-t)=f(t) or f(-t)=-f(t) depending on whether you're in the odd or even case to investigate how F(-x) relates to F(x).
 
  • #8
5,598
213
Yes you did.



Given a function f(x) that is integrable on the appropriate intervals define:
[tex]F(x) = \int_0^x f(t) dt[/tex]
Then you wish to show that f(x) is odd imply F(x) even, and f(x) even imply F(x) odd. In general we have the identity:

[tex]\int_a^b f(t) dt = -\int_{-a}^{-b}f(-t)dt[/tex]
You can use that to write:
[tex]F(-x) = \int_0^{-x} f(t)dt = -\int_0^x f(-t) dt[/tex]
Now you can use f(-t)=f(t) or f(-t)=-f(t) depending on whether you're in the odd or even case to investigate how F(-x) relates to F(x).

Thanks for your help, I need to verify again.


Let u=-t, therefore du=-dt. Also the integration of t from -a to -b imply integration of u from a to b.

[tex]\Rightarrow -\int_{-a}^{-b}f(-t)dt = \int_{a}^{b}f(u)du = F(b)-F(a)[/tex]


[tex]\int_{a}^{b}f(t)dt = F(b)-F(a)\Rightarrow \int_a^b f(t) dt = -\int_{-a}^{-b}f(-t)dt = F(b)-F(a)[/tex]

[tex]\Rightarrow F(x)=\int_0^x f(t)dt = -\int _0^{-x}f(-t)dt[/tex]

If f(x) is even, f(-t)=f(t):

[tex]\Rightarrow F(x)=\int_0^x f(t)dt = -\int _0^{-x}f(-t)dt = -\int _0^{-x}f(t)dt = -[F(-x)-F(0)] = -F(-x)[/tex]

This give F(x)=-F(-x) which imply if f(x) is even, F(x) is odd. Therefore integration of an even function gives an odd function.

There other way would be the same that integrate of an odd function give an even function.

Am I correct?
 

Related Threads on Prove derivative of an odd function is even and vise versa.

Replies
2
Views
2K
Replies
12
Views
8K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
7
Views
3K
Replies
1
Views
658
Replies
10
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
18
Views
13K
  • Last Post
Replies
4
Views
1K
Top