A function \(f=u+iv\) on an open set \(\Omega\subseteq\CC\) is holomorphic if \(f'(z)=\lim_{h\to0}\frac{f(z+h)-f(z)}{h}\) exists at every \(z\in\Omega\). Taking the limit along real and along imaginary \(h\) gives the Cauchy–Riemann equations \(u_x=v_y\), \(u_y=-v_x\), equivalently \(\partial f/\partial\bar z=0\); conversely, if \(u\) and \(v\) are continuously differentiable and satisfy these equations, then \(f\) is holomorphic. A function is analytic on \(\Omega\) if it is given by a convergent power series on a neighbourhood of each point. A power series is holomorphic on its disc of convergence, with derivative the termwise derivative, so analytic functions are holomorphic; the converse follows from the Cauchy integral formula (Cauchy theory). If \(u\) and \(v\) are twice continuously differentiable, the Cauchy–Riemann equations give \(\Delta u=\Delta v=0\).
Holomorphic and entire functions (Tag D-E7A5W)
Let \(\Omega\subseteq\CC\) be open, let \(f\colon\Omega\to\CC\), and let \(z_0\in\Omega\). The function \(f\) is complex differentiable, or holomorphic, at \(z_0\) if the limit \[ f'(z_0)\coloneqq\lim_{\substack{h\to0\\ h\in\CC\setminus\{0\}}}\frac{f(z_0+h)-f(z_0)}{h} \] exists. The function \(f\) is holomorphic on \(\Omega\) if it is holomorphic at every point of \(\Omega\), and a function \(f\colon\CC\to\CC\) that is holomorphic on \(\CC\) is entire.
Let \(\Omega\subseteq\CC\) be open, \(f\colon\Omega\to\CC\), and \(z_0\in\Omega\). Then \(f\) is holomorphic at \(z_0\) if and only if there exist \(\alpha\in\CC\) and a function \(\psi\), defined for \(h\in\CC\) with \(z_0+h\in\Omega\), such that \(\psi(h)\to0\) as \(h\to0\) and \[ f(z_0+h)-f(z_0)=\alpha h+h\,\psi(h). \] In this case \(\alpha=f'(z_0)\).
If \(f'(z_0)\) exists, put \(\alpha\coloneqq f'(z_0)\), \(\psi(0)\coloneqq0\), and \(\psi(h)\coloneqq\frac{f(z_0+h)-f(z_0)}{h}-\alpha\) for \(h\neq0\); then \(\psi(h)\to0\) and the identity holds. Conversely, dividing the identity by \(h\neq0\) gives \(\frac{f(z_0+h)-f(z_0)}{h}=\alpha+\psi(h)\to\alpha\), so \(f'(z_0)\) exists and equals \(\alpha\).
See [@SS03].
Complex differentiable implies Cauchy--Riemann (Tag PR-37QA5)
Let \(\Omega\subseteq\CC\) be open, let \(f=u+iv\colon\Omega\to\CC\) with \(u,v\) real-valued, and regard \(f\), \(u\), \(v\) as functions of \((x,y)\) where \(z=x+iy\). If \(f\) is complex differentiable at \(z_0=x_0+iy_0\in\Omega\), then the partial derivatives of \(u\) and \(v\) exist at \((x_0,y_0)\), \[ f'(z_0)=\dd{f}{x}(x_0,y_0)={1\over i}\dd{f}{y}(x_0,y_0), \] and at \((x_0,y_0)\) \[ \dd{u}{x} = \dd{v}{y}, \qquad \dd{u}{y} = -\dd{v}{x}. \]
The limit \(f'(z_0)=\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h}\) exists, so it may be computed along any direction. Along real \(h=h_1\in\RR\), \[ f'(z_0) = \lim_{h_1\to 0} { f(x_0+h_1, y_0) - f(x_0, y_0) \over h_1} = \dd{f}{x}(x_0, y_0). \] Along purely imaginary \(h=ih_2\) with \(h_2\in\RR\), \[ f'(z_0) = \lim_{h_2\to 0} { f(x_0, y_0+h_2) - f(x_0, y_0) \over ih_2 } = {1\over i} \dd{f}{y}(x_0, y_0). \] Taking real and imaginary parts of these limits shows that the partial derivatives of \(u\) and \(v\) exist. Writing \(1/i=-i\), \[ \begin{aligned} \dd{f}{x} &= \dd{u}{x} + i \dd{v}{x}, \\ {1\over i} \dd{f}{y} &= -i \qty{ \dd{u}{y} + i \dd{v}{y}} = \dd{v}{y} - i\dd{u}{y}. \end{aligned} \] Equating real and imaginary parts of the two expressions for \(f'(z_0)\) gives \(\dd{u}{x} = \dd{v}{y}\) and \(\dd{v}{x} = -\dd{u}{y}\).
Cauchy--Riemann implies holomorphic (Tag PR-JKE6C)
Let \(\Omega\subseteq\CC\) be open and let \(f = u+iv\colon\Omega\to\CC\), where \(u,v\colon\Omega\to\RR\) are continuously differentiable as functions of \((x,y)\), \(z=x+iy\). If \(u_x=v_y\) and \(u_y=-v_x\) on \(\Omega\), then \(f\) is holomorphic on \(\Omega\) and \[ f'(z) = \del f(z) = u_x(z) + iv_x(z). \]
Fix \(z\in\Omega\) and write \(h=h_1+ih_2\). Since \(u\) and \(v\) are continuously differentiable, they are real differentiable at \(z\), so \[ f(z+h)-f(z)=u_xh_1+u_yh_2+i\qty{v_xh_1+v_yh_2}+o(\abs{h}), \] with the partial derivatives evaluated at \(z\). Substituting \(u_y=-v_x\) and \(v_y=u_x\) gives \(u_xh_1-v_xh_2+i\qty{v_xh_1+u_xh_2}=(u_x+iv_x)(h_1+ih_2)\), so \(\frac{f(z+h)-f(z)}{h}\to u_x+iv_x\) as \(h\to0\). Finally, \(\del f=\frac12\qty{f_x-if_y}=\frac12\qty{u_x+v_y+i(v_x-u_y)}=u_x+iv_x\) by the same equations.
Wirtinger operators $\partial$ and $\bar\partial$ (Tag D-OAFF5)
Let \(U\subseteq\CC\) be open, identify \(\CC\) with \(\RR^2\) by \(z=x+iy\), and let \(F\colon U\to\CC\) be \(C^1\) as a function of \((x,y)\). The Wirtinger derivatives of \(F\) are \[ \frac{\partial F}{\partial z}\coloneqq\frac12\qty{\frac{\partial F}{\partial x}-i\frac{\partial F}{\partial y}},\qquad \frac{\partial F}{\partial\bar z}\coloneqq\frac12\qty{\frac{\partial F}{\partial x}+i\frac{\partial F}{\partial y}}, \] and the operators \(\del\coloneqq\del_z\coloneqq\frac12(\del_x-i\del_y)\) and \(\delbar\coloneqq\del_{\bar z}\coloneqq\frac12(\del_x+i\del_y)\) are the del and del-bar operators.
Since \(\frac1i=-i\), the same formulas read \(\frac{\partial F}{\partial z}=\frac12\qty{\frac{\partial F}{\partial x}+\frac1i\frac{\partial F}{\partial y}}\) and \(\frac{\partial F}{\partial\bar z}=\frac12\qty{\frac{\partial F}{\partial x}-\frac1i\frac{\partial F}{\partial y}}\).
With \(F\) as in the definition, \(dz=dx+i\,dy\), and \(d\bar z=dx-i\,dy\), the differential of \(F\) is \[ dF=\frac{\partial F}{\partial z}\dz+\frac{\partial F}{\partial\bar z}\dzbar. \]
Substituting \(dz\) and \(d\bar z\), the right side equals \[ \frac12\qty{F_x-iF_y}(dx+i\,dy)+\frac12\qty{F_x+iF_y}(dx-i\,dy)=F_x\dx+F_y\dy=dF. \]
Holomorphic at a point if and only if $\bar\partial f$ vanishes there (Tag PR-TVPCM)
Let \(f=u+iv\) be defined on a neighborhood of \(z_0\in\CC\), with \(u,v\) real-valued and real-differentiable at \(z_0\). Then \(f\) is holomorphic at \(z_0\) if and only if \[ \delbar f(z_0)=\frac12(\del_x+i\del_y)f(z_0)=0. \]
At \(z_0\), \[ \begin{aligned} 2\delbar f&=(\del_x+i\del_y)(u+iv)\\ &=u_x+iv_x+iu_y-v_y\\ &=(u_x-v_y)+i(u_y+v_x). \end{aligned} \] So \(\delbar f(z_0)=0\) exactly when \(u_x=v_y\) and \(u_y=-v_x\) at \(z_0\), the Cauchy–Riemann equations. For \(f\) real-differentiable at \(z_0\), these hold if and only if the real derivative of \(f\) at \(z_0\) is multiplication by a complex number, that is, if and only if \(f\) is complex differentiable at \(z_0\).
Laplacian and harmonic functions (Tag D-CFBSA)
Let \(\Omega\subseteq\RR^2\) be open and let \(u\colon\Omega\to\RR\) be a \(C^2\) function of \((x,y)\). The Laplacian of \(u\) is \[ \Delta u\coloneqq\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}, \] and \(u\) is harmonic on \(\Omega\) if \(\Delta u=0\) on \(\Omega\).
Holomorphic functions have harmonic components (Tag PR-K57J6)
Let \(\Omega\subseteq\CC\) be open and let \(f(z) = u(x, y) + iv(x, y)\) be holomorphic on \(\Omega\), with \(u,v\) real-valued and \(z=x+iy\). Then \(u\) and \(v\) are harmonic on \(\Omega\).
The derivative of a holomorphic function is holomorphic, so \(f\) is infinitely differentiable and \(u,v\) have continuous partial derivatives of all orders. By the Cauchy–Riemann equations \(u_x=v_y\) and \(u_y=-v_x\), so \(u_{xx}=v_{yx}=v_{xy}=-u_{yy}\) and \(v_{xx}=-u_{yx}=-u_{xy}=-v_{yy}\), using equality of mixed partials. Hence \(\Delta u=\Delta v=0\).
Analytic function (Tag D-V6UQJ)
Let \(\Omega\subseteq\CC\) be open, let \(f\colon\Omega\to\CC\), and let \(z_0\in\Omega\). The function \(f\) is analytic at \(z_0\) if there exist a power series \(\sum_{n\ge0}a_n(z-z_0)^n\) with radius of convergence \(R>0\) and an open neighborhood \(U\subseteq\Omega\) of \(z_0\) with \(\abs{z-z_0}<R\) for all \(z\in U\), such that \[ f(z)=\sum_{n\ge0}a_n(z-z_0)^n\qquad\text{for all } z\in U. \] The function \(f\) is analytic on \(\Omega\) if it is analytic at every point of \(\Omega\).
Power series are smooth on their disc of convergence (Tag PR-QNDSD)
Let \(f(z)=\sum_{k\ge0}c_k(z-z_0)^k\) be a power series with radius of convergence \(R>0\). Then \(f\) is holomorphic and infinitely differentiable on \(D_R(z_0)\), its derivative is obtained by term-by-term differentiation, \[ f'(z)=\sum_{k\ge1}kc_k(z-z_0)^{k-1},\qquad z\in D_R(z_0), \] where this series also has radius of convergence \(R\), and the coefficients are \[ c_k=\frac{f^{(k)}(z_0)}{k!},\qquad k\ge0. \]