Project ForgeEngineering Mathematics
Calculus
GATE 2023
Hard
The value of k that makes the complex-valued function f(z)=e^(−kx)(cos2y − i·sin2y) analytic, where z=x+iy, is ______.
2(± 0.05 accepted)
Explanation
With u=e^(−kx)cos2y and v=−e^(−kx)sin2y, the Cauchy-Riemann equations require ∂u/∂x=∂v/∂y and ∂u/∂y=−∂v/∂x. The first gives −k·e^(−kx)cos2y=−2e^(−kx)cos2y, so k=2. The second gives the same result, confirming k=2.
Source: GATE ME 2023 Official Paper (2023)
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