“Cauchy-Riemann Equations”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cauchy-Riemann Equations

释义

Cauchy-Riemann Equations

Let

(1)

where

(2)

(3)

The total derivative of

with respect to

may then be computed as follows.

			(4)
			(5)

			(6)
			(7)

and

(8)

In terms of

and

, (8) becomes


			(9)

Along the real, or x-Axis,

, so

(10)

Along the imaginary, or

-axis,

, so

(11)

is Complex Differentiable, then the value of the derivative must be the same for a given

, regardless of itsorientation. Therefore, (10) must equal (11), which requires that

(12)

and

(13)

These are known as the Cauchy-Riemann equations. They lead to the condition

(14)

The Cauchy-Riemann equations may be concisely written as


			(15)

In Polar Coordinates,

(16)

so the Cauchy-Riemann equations become

			(17)
			(18)

and

satisfy the Cauchy-Riemann equations, they also satisfyLaplace's Equation in 2-D, since

(19)

(20)

By picking an arbitrary

, solutions can be found which automatically satisfy the Cauchy-Riemann equations andLaplace's Equation. This fact is used to find so-called Conformal Solutions tophysical problems involving scalar potentials such as fluid flow and electrostatics.

See also Cauchy Integral Theorem, Conformal Solution, Monogenic Function,Polygenic Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 17, 1972.

Arfken, G. ``Cauchy-Riemann Conditions.'' §6.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 3560-365, 1985.

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