exsheet6 - Sivakumar M407 Example Sheet 6 1 Let a< b be...

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Unformatted text preview: Sivakumar M407 Example Sheet 6 1. Let a < b be real numbers, and suppose that G,H : [ a,b ] → C are continuous on [ a,b ]. Let α be a fixed complex number. Verify the following statements: (i) b R a ( G ( t ) + H ( t )) dt = b R a G ( t ) dt + b R a H ( t ) dt . (ii) b R a ( αG )( t ) dt = α b R a G ( t ) dt . 2. Suppose that γ is a smooth arc parametrized by z ( t ), a ≤ t ≤ b . Let φ : [ c,d ] → [ a,b ] be a function satisfying the following conditions: (i) φ is continuously differentiable on [ c,d ], (ii) φ ( s ) > 0 for every c < s < d , and (iii) φ ( c ) = a and φ ( d ) = b . (i) Verify that the function w : [ c,d ] → C given by w ( s ) := z ( φ ( s )), c ≤ s ≤ d , is continuously differentiable on [ c,d ]. (ii) Verify that the function w also parametrizes γ . (iii) Let f : γ → C be continuous on γ . Show that d R c f ( w ( s )) w ( s ) ds = b R a f ( z ( t )) z ( t ) dt . (This shows that the line integral of f over γ is invariant under a smooth change of parameter.)is invariant under a smooth change of parameter....
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exsheet6 - Sivakumar M407 Example Sheet 6 1 Let a< b be...

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