A holomorphic function is a function that is differentiable at every point on its domain. In order for it to be differentiable, it needs to satisfy the Cauchy-Riemann equation properties, such that:
f(z) = u(x,y) + iv(x,y)
ux = vy
vx = -uy
If that is so, then f'(z) = ux + ivx
Otherwise, if a function doesn't satisfy these conditions, we say that it's not holomorphic. For instance:
f(z) = z̅
Test with the following properties:
ux = vy
vx = -uy
z̅ is written as u(x,y) - iv(x,y). Take the partial derivatives of u(x,y) and v(x,y). Then:
ux = -vy
vx = -(-uy) = uy
Since the conditions don't hold, that function is not holomorphic.
The parent function of the exponential function is ax
A __________ function takes the exponential function's output and returns the exponential function's input.
No. The inverse of an exponential function is a logarithmic function.
Logarithmic Function
An equation where the left is the function of the right. f(x)=x+3 is function notation. The answer is a function of what x is. f(g(x))= the answer the inside function substituted in the outside function.
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The Cauchy-Riemann equations are fundamental in complex analysis, providing conditions for a function to be holomorphic, meaning it is complex differentiable. These equations are essential in various fields, including fluid dynamics, where they describe potential flow, and in electrical engineering for analyzing electromagnetic fields. Additionally, they are used in conformal mapping, which allows for the transformation of complex shapes in a way that preserves angles, facilitating the solution of physical problems in engineering and physics.
Function
The parent function of the exponential function is ax