1)If then and . That means .
Now if then . With . Then
The problem if that if there existed an entire function such that then the two integrals must be the same because the Fundamental Theorem of Line Integrals implies path independence.
Can you help with the attached problem? I think for part (i) and (ii) the answer is the same - i have this as 0 in both cases. However, i'm having problems with part (a) (iii). How do i prove that its not an entire function. Has this got anything to do with the Grid Path Theorem?
For part (b) i have an answer of (3pi/26)(9+e^6). For some reason, this looks incorrect. Can someone work through the problem and tell me if this is what they also get?
For part (c) (i) I get a final answer of -e^3 + e^-3. Is this right and if so does it simplify into a trigonometric function?
For part (c) (ii) I get a final answer of 0. I have done the following for this:
f(z)=zcosh(z^2) = 0.5(2z)cosh(z^2). Hence a primitive of f is f(z) = 0.5sinh(z^2) and the values of 3 and -3 are input into this and subtracted from each other to give 0. Is this correct?