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Unformatted text preview: x )(sin y )-e-x (sin y + y cos y ) + C ( y ) v = e-x (-x sin y + y cos y ) (4) Clearly ze-z is analytic since z ,-z and e z are analytic, and compositions and products analytic functions are analytic. Since u = Re f , we know u is harmonic. (5) | e iz + e-iz | ≤ | e iz | + | e-iz | = | e ix e-y | + | e-ix e y | = | e-y | + | e y | (6) Branch cut is at π 2 (-1) i = e i log(-1) = e i log( e i ( π +2 πk ) ) = e i (ln 1+ i ( π )) = e-π + i (7) sin( iz ) = e i ( iz )-e-i ( iz ) 2 i = e-z-e z 2 i = i e z-e-z 2 = i sinh( z ) 2...
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This note was uploaded on 05/11/2008 for the course MAT 461 taught by Professor Ibrahim during the Fall '07 term at ASU.
- Fall '07