Differentiability

Any differentiable complex function F(z) is the complex potential for a 2D incompressible potential flow. Further is the complex conjugate velocity.

Reason: differentiability requires that is the same whichever direction we take . In particular if we take , a change in x only, we should get the same as when we take , a change in y only. That means, if :

Since ,

from which it follows that u and v satisfy both the continuity equation and irrotationality:

Differentiable complex functions are easy to find:

• Constants are differentiable: .
• Function F(z)=z is differentiable: .
• Sums of differentiable functions are differentiable (see your calculus book.)
• Products of differentiable functions are differentiable (see your calculus book.)
• Inverses of differentiable functions are differentiable (see your calculus book.)
• Functions with converging Taylor series are differentiable.

Not differentiable:

• Purely real functions such as x, , ...
• Purely imaginary functions such as , , ...
• Complex conjugates such as .

In other words, taking real or imaginary parts, absolute values, complex conjugates, ..., all make the expression nondifferentiable.

Exercise:

Give as many 2d incompressible potential flows as you can.

You should now be able to do 18.2, 3