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futurebird
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I'm studying the proof of the [tex]ML[/tex] inequlity from complex analysis. I don't know what they did in one step of the proof and I was wondering if anyone can explain the step to me.
First of all the theorem says:
Let f(z) be continuous on a contour C. Then
[tex]|\int_{c}f(z)dz| \leq ML[/tex]
Where [tex]L[/tex] is the length of C and [tex]M[/tex] is an upper bound for |f| on C.
PROOF:
After a few steps we have:
[tex]|\int_{c}f(z)dz| \leq \int^{b}_{a} |f(z(t))||z'(t)|dt[/tex]
And the text says " |f| is bounded on C, [tex]|f(z)| \leq M[/tex] on C, where [tex]M[/tex] is constant, then:
[tex]|\int_{c}f(z)dz| \leq M\int^{b}_{a} |z'(t)|dt[/tex]
How did they get the [tex]M[/tex] out of the integral?
First of all the theorem says:
Let f(z) be continuous on a contour C. Then
[tex]|\int_{c}f(z)dz| \leq ML[/tex]
Where [tex]L[/tex] is the length of C and [tex]M[/tex] is an upper bound for |f| on C.
PROOF:
After a few steps we have:
[tex]|\int_{c}f(z)dz| \leq \int^{b}_{a} |f(z(t))||z'(t)|dt[/tex]
And the text says " |f| is bounded on C, [tex]|f(z)| \leq M[/tex] on C, where [tex]M[/tex] is constant, then:
[tex]|\int_{c}f(z)dz| \leq M\int^{b}_{a} |z'(t)|dt[/tex]
How did they get the [tex]M[/tex] out of the integral?
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