Editing Cauchy–Riemann equations (section)

=== Complex differentiability ===
Let
<math display="block"> f(z) = u(z) + i \cdot v(z) </math>
where <math display="inline">u</math> and <math>v</math> are [[Real-valued function|real-valued functions]], be a [[complex-valued function]] of a complex variable <math display="inline"> z = x + i y</math> where <math display="inline"> x</math> and <math display="inline"> y</math> are real variables. <math display="inline"> f(z) = f(x + iy) = f(x,y)</math> so the function can also be regarded as a function of real variables <math display="inline">x</math> and <math display="inline"> y</math>. Then, the ''complex-derivative'' of <math display="inline"> f </math> at a point <math display="inline"> z_0=x_0+iy_0 </math> is defined by
<math display="block"> f'(z_0) =\lim_{\underset{h\in\Complex}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} </math>
provided this limit exists (that is, the limit exists along every path approaching <math display="inline"> z_{0} </math>, and does not depend on the chosen path).

A fundamental result of [[complex analysis]] is that <math>f</math> is [[differentiable function#Differentiability in complex analysis|complex differentiable]] at <math>z_0</math> (that is, it has a complex-derivative), [[if and only if]] the bivariate [[real function]]s <math>u(x+iy)</math> and <math>v(x+iy)</math> are [[differentiable]] at <math>(x_0,y_0),</math> and satisfy the Cauchy–Riemann equations at this point.{{sfn|Rudin|1966}}{{sfn|Marsden|Hoffman|1973}}<ref>{{cite book|first=A.I.|last=Markushevich|title=Theory of functions of a complex variable 1|publisher=Chelsea|year=1977}}, p. 110-112 (Translated from Russian)</ref>

In fact, if the complex derivative exists at <math display="inline"> z_0</math>, then it may be computed by taking the limit at <math display="inline"> z_0</math> along the real axis and the imaginary axis, and the two limits must be equal. Along the real axis, the limit is 
<math display="block">\lim_{\underset{h\in\Reals}{h\to 0}} \frac{f(z_0+h)-f(z_0)}{h} = \left.  \frac{\partial f}{\partial x} \right \vert_{z_0}</math>
and along the imaginary axis, the limit is
<math display="block">\lim_{\underset{h\in \Reals}{h\to 0}} \frac{f(z_0+ih)-f(z_0)}{ih} = \left. \frac{1}{i}\frac{\partial f}{\partial y} \right \vert _{z_0}.</math>

So, the equality of the derivatives implies 
<math display="block">i \left. \frac{\partial f}{\partial x} \right \vert _{z_0} = \left. \frac{\partial f}{\partial y} \right \vert _{z_0}</math>
which is the complex form of Cauchy–Riemann equations ({{EquationNote|2}}) at <math display="inline"> z_0</math>.

(Note that if <math>f</math> is complex differentiable at <math>z_0</math>, it is also real differentiable and the [[Jacobian]] of <math>f</math> at <math>z_0</math> is the complex scalar <math>f'(z_0)</math>, regarded as a real-linear map of <math>\mathbb C</math>, since the limit <math>|f(z)-f(z_0)-f'(z_0)(z-z_0)|/|z-z_0|\to 0</math> as <math>z\to z_0</math>.)

Conversely, if {{mvar|f}} is differentiable at <math display="inline"> z_{0} </math> (in the real sense) and satisfies the Cauchy-Riemann equations there, then it is complex-differentiable at this point. Assume that {{mvar|f}} as a function of two real variables {{mvar|x}} and {{mvar|y}} is differentiable at {{math|''z''<sub>0</sub>}} (real differentiable). This is equivalent to the existence of the following linear approximation <math display="block"> \Delta f(z_0) = f(z_0 + \Delta z) - f(z_0) = f_x \,\Delta x + f_y \,\Delta y + \eta(\Delta z)</math>where <math display="inline"> f_x = \left. \frac{\partial f}{\partial x}\right \vert _{z_0} </math>, <math display="inline"> f_y = \left. \frac{\partial f}{\partial y} \right \vert _{z_0} </math>, {{math|1=''z'' = ''x'' + ''iy''}}, and <math display="inline">\eta(\Delta z) / |\Delta z| \to 0</math> as {{math|Δ''z'' → 0}}. 

Since <math display="inline"> \Delta z + \Delta \bar{z}= 2 \, \Delta x </math> and <math display="inline"> \Delta z - \Delta \bar{z}=2i \, \Delta y </math>, the above can be re-written as

<math display="block"> \Delta f(z_0) = \frac{f_x - if_y}{2} \, \Delta z + \frac{f_x + if_y}{2} \, \Delta \bar{z} + \eta(\Delta z)\, </math><math display="block">\frac{\Delta f}{\Delta z} = \frac{f_x -i f_y}{2}+ \frac{f_x + i f_y}{2}\cdot \frac{\Delta\bar{z}}{\Delta z} + \frac{\eta(\Delta z)}{\Delta z}, \;\;\;\;(\Delta z \neq 0). </math>  

Now, if <math display="inline">\Delta z</math> is real, <math display="inline">\Delta\bar z/\Delta z = 1</math>, while if it is imaginary, then <math display="inline">\Delta\bar z/\Delta z=-1</math>. Therefore, the second term is independent of the path of the limit <math display="inline">\Delta z\to 0</math> when (and only when) it vanishes identically: <math display="inline">f_x + i f_y=0</math>, which is precisely the Cauchy–Riemann equations in the complex form. This proof also shows that, in that case,
<math display="block">\left.\frac{df}{dz}\right|_{z_0} = \lim_{\Delta z\to 0}\frac{\Delta f}{\Delta z} = \frac{f_x - i f_y}{2}.</math>

Note that the hypothesis of real differentiability at the point <math>z_0</math> is essential and cannot be dispensed with. For example,<ref>{{cite book|first=E|last=Titchmarsh|title=The theory of functions|year=1939|publisher=Oxford University Press}}, 2.14</ref> the function <math>f(x,y) = \sqrt{|xy|}</math>, regarded as a complex function with imaginary part identically zero, has both partial derivatives at <math>(x_0,y_0)=(0,0)</math>, and it moreover satisfies the Cauchy–Riemann equations at that point, but it is not differentiable in the sense of real functions (of several variables), and so the first condition, that of real differentiability, is not met. Therefore, this function is not complex differentiable.

Some sources<ref>{{Cite book |last1=Arfken |first1=George B. |title=Mathematical Methods for Physicists: A Comprehensive Guide |last2=Weber |first2=Hans J. |last3=Harris |first3=Frank E. |publisher=Academic Press |year=2013 |isbn=978-0-12-384654-9 |edition=7th |pages=471–472 |language=English |chapter=11.2 CAUCHY-RIEMANN CONDITIONS}}
</ref><ref>{{Cite book |last=Hassani |first=Sadri |title=Mathematical Physics: A Modern Introduction to Its Foundations |publisher=Springer |year=2013 |isbn=978-3-319-01195-0 |edition=2nd |pages=300–301 |language=English |chapter=10.2 Analytic Functions}}</ref> state a sufficient condition for the complex differentiability at a point <math>z_0</math> as, in addition to the Cauchy–Riemann equations, the partial derivatives of <math>u</math> and <math>v</math> be continuous at the point because this continuity condition ensures the existence of the aforementioned linear approximation. Note that it is not a necessary condition for the complex differentiability. For example, the function <math>f(z) = z^2e^{i/|z|}</math> is complex differentiable at 0, but its real and imaginary parts have discontinuous partial derivatives there. Since complex differentiability is usually considered in an open set, where it in fact implies continuity of all partial derivatives (see [[#Goursat's theorem and its generalizations|below]]), this distinction is often elided in the literature.