For what constant value (or values) of a is the function u = x + ax3 t

1.	For what constant value (or values) of a is the function u = x + ax3 the real part of an analytic function? Find the imaginary part of this analytic function. Express this analytic function as a function of z.
Answer» u_xx = 6ax , u_yy = 0 . Thus ∇²u = 0 only if a = 0, in which case u = x. The conjugate of x, denoted as v, satisfies v_y = u_x = 1 and v_x = −u_y = 0. We get v = y + c, and u + iv = z + ic, where c is a constant.

For what constant value (or values) of a is the function u = x + ax3 the real part of an analytic function? Find the imaginary part of this analytic function. Express this analytic function as a function of z.

Answer»

u_xx = 6ax , u_yy = 0 .

Thus ∇²u = 0 only if a = 0, in which case

u = x.

The conjugate of x, denoted as v, satisfies

v_y = u_x = 1 and v_x = −u_y = 0.

We get

v = y + c, and u + iv = z + ic,

where c is a constant.

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For what constant value (or values) of a is the function u = x + ax3 the real part of an analytic function? Find the imaginary part of this analytic function. Express this analytic function as a function of z.

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