1-复变函数 - 图文

例：试推导极坐标下的C.R.方程

方法一：从极坐标关系出发，分别考虑?z

沿径向和沿横向趋于零。

f(z)?u(?,?)?iv(?,?)z??ei??z???e??ei??i?i??沿径向趋于零，即???0,???0??e?u(????,?)?u(?,?)??i?v(????,?)?v(?,?)??limi????0??e???0f'??z??lim?u(????,?)?iv(????,?)???u(?,?)?iv(?,?)?i?1?i?e??u?v??????i???????沿横向趋于零，即???0,???0?ei???u(?,????)?u(?,?)??i?v(?,????)?v(?,?)??limi????0?ei?????0f'??z??lim?u(?,????)?iv(?,????)???u(?,?)?iv(?,?)?i?1??u?v?1??u?v????i???i?i???i??????ei???????e???????f(z)=u(ρ,φ)+iv(ρ,φ)

在z点可导须两极限相等

??u1?v?????????1?u?v??????????30

方法二：从直角坐标关系出发

f(z)?u(x,y)?iv(x,y)?x??cos?,??y??sin??u?u?x?u?y?u?u?v?v???cos??sin??cos??sin????x???y???x?y?y?x?v?v?x?v?y?v?v?u?u???cos??sin???cos??sin????x???y???x?y?y?x??u??u?u?x?u?y?u?u?u???(??sin?)??cos??????cos??sin???????x???y???x?y?y?x????v??v?v?x?v?y?v?v?v????(??sin?)??cos????cos??sin???y????x???y???x?y?x??本节作业：1.第12页习题

2.复习静电学中“电场强度与电势梯度的关系”部分。

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