Integration with variables Notes_Part_4

Integration with variables Notes_Part_4 - The proof of the...

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The proof of the converse — that any function whose real and imaginary components satisfy the Cauchy–Riemann equations is diFerentiable — will be omitted, but can be found in any basic text on complex analysis, e.g., [ 3 , 65 , 118 ]. Remark : It is worth pointing out that equation (7.19) tells us that f satis±es ∂f/∂x = i ∂f/∂y , which, reassuringly, agrees with the ±rst equation in (7.5). Example 7.4. Consider the elementary function z 3 = ( x 3 3 xy 2 ) + i (3 x 2 y y 3 ) . Its real part u = x 3 3 xy 2 and imaginary part v = 3 x 2 y y 3 satisfy the Cauchy–Riemann equations (7.18), since ∂u ∂x = 3 x 2 3 y 2 = ∂v ∂y , ∂u ∂y = 6 xy = ∂v ∂x . Theorem 7.3 implies that f ( z ) = z 3 is complex diFerentiable. Not surprisingly, its deriva- tive turns out to be f ( z ) = ∂u ∂x + i ∂v ∂x = ∂v ∂y i ∂u ∂y = (3 x 2 3 y 2 ) + i (6 xy ) = 3 z 2 . ²ortunately, the complex derivative obeys all of the usual rules that you learned in real-variable calculus. ²or example, d dz z n = n z n 1 , d dz e cz = c e cz , d dz log z = 1 z , (7 . 20) and so on. The power n can be non-integral — or even, in view of the identity z n = e n log z , complex, while c is any complex constant. The exponential formulae (7.14) for the complex trigonometric functions implies that they also satisfy the standard rules d dz cos z = sin z, d dz sin z = cos z. (7 . 21) The formulae for diFerentiating sums, products, ratios, inverses, and compositions of com- plex functions are all identical to their real counterparts, with similar proofs. Thus, thank- fully, you don’t need to learn any new rules for performing complex diFerentiation! Remark
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Integration with variables Notes_Part_4 - The proof of the...

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