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Diffstat (limited to 'ClassJune26.page')
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diff --git a/ClassJune26.page b/ClassJune26.page index 5267204..fb826d9 100644 --- a/ClassJune26.page +++ b/ClassJune26.page @@ -189,10 +189,8 @@ curve $\gamma$ through $z$ to a smooth curve $f\circ\gamma$ through $f(z)$. What happens to the tangent of $\gamma$ at $z$? Given by the derivative $df(z)$, a linear map taking vectors based at $z$ to vectors based at $f(z)$. If we use rectangular coordinates - $z\mapsto f(z)$ - - $x+iy\mapsto u(x,y)+iv(x,y)$ - + $z\mapsto f(z)$ + $x+iy \mapsto u(x,y)+iv(x,y)$ $\left(\begin{array}{c} x\\ y\end{array}\right)\mapsto\left(\begin{array}{c} @@ -231,7 +229,7 @@ ie, it looks just like multiplication by the complex number $a+bi$. The function $f$ is conformal if its derivative acts like a nonzero complex number. Analytically, this condition is given by the following differential equations, called the **Cauchy-Riemann equations**: -$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\mbox{ and }\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.$ +$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\mbox{ and }\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}.$ A complex function $f=u+iv$ is said to be **holomorphic** if $f$ satisfies the CR. We've shown that conformal $\Longrightarrow$ holomorphic. Holomorphic functions are slightly more general, as the |