The Cauchy–Riemann equations

Theorem

The Cauchy–Riemann equations

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Statement

Let \( \Omega \subseteq \mathbb{C} \) be open, let \( z_0 = x_0+iy_0 \in \Omega \), and write \( f:\Omega\to\mathbb{C} \) as \( f(x+iy) = u(x,y)+iv(x,y) \) with \( u,v:\Omega\to\mathbb{R} \) (identifying \( \Omega \subseteq \mathbb{C} \) with an open subset of \( \mathbb{R}^2 \)). (Necessity.) If \( f \) is complex differentiable at \( z_0 \), i.e. \( f'(z_0) = \lim_{h\to 0}\frac{f(z_0+h)-f(z_0)}{h} \) exists (\( h\in\mathbb{C} \)), then the partial derivatives \( u_x,u_y,v_x,v_y \) exist at \( (x_0,y_0) \) and satisfy the Cauchy–Riemann equations \[ u_x(x_0,y_0) = v_y(x_0,y_0), \qquad u_y(x_0,y_0) = -v_x(x_0,y_0), \] and moreover \( f'(z_0) = u_x(x_0,y_0)+iv_x(x_0,y_0) \). (Sufficiency.) Conversely, if \( u,v \) have partial derivatives \( u_x,u_y,v_x,v_y \) throughout some open neighbourhood of \( (x_0,y_0) \), these partials are continuous at \( (x_0,y_0) \), and the Cauchy–Riemann equations hold at \( (x_0,y_0) \), then \( f \) is complex differentiable at \( z_0 \), with \( f'(z_0)=u_x(x_0,y_0)+iv_x(x_0,y_0) \).

Why it matters

The Cauchy–Riemann (CR) equations are the bridge between the algebraic notion of complex differentiability — a single complex limit — and the analytic machinery of two-variable real calculus. They convert "does \( f'(z_0) \) exist?" into a checkable system of two coupled first-order PDEs relating the real and imaginary parts of \( f \), which is exactly what makes the theory of holomorphic functions computationally tractable: one can verify analyticity of an explicit map by differentiating its real and imaginary parts as ordinary two-variable functions.

The equations are also the seed of everything downstream in complex analysis: they force \( u \) and \( v \) to be harmonic (once \( f \) is known to be holomorphic and hence \( C^\infty \), via Goursat/Cauchy's theorem), they encode the fact that \( f \) acts locally as a rotation-and-scaling (conformality) rather than a general linear map, and they are the two-dimensional shadow of the much stronger rigidity that distinguishes holomorphic functions from merely smooth real maps \( \mathbb{R}^2\to\mathbb{R}^2 \).

Hypotheses

\( \Omega \) open, \( f \) defined in a full neighbourhood of \( z_0 \). Complex differentiability is a two-sided limit over all directions \( h\to 0 \) in \( \mathbb{C} \); if \( f \) were only defined on, say, a ray or the real axis, "\( \lim_{h\to0} \frac{f(z_0+h)-f(z_0)}{h} \)" would not test enough directions and the CR equations would carry no content (e.g. \( f(x)=x \) restricted to \( \mathbb{R} \) trivially "satisfies" one-dimensional differentiability without CR being meaningful).

(Sufficiency direction) Partials continuous at \( (x_0,y_0) \), not merely existing there. Existence of \( u_x,u_y,v_x,v_y \) at the single point \( (x_0,y_0) \) together with CR is not enough: take \( f(z) = \dfrac{\bar z^{\,2}}{z} \) for \( z\neq0 \), \( f(0)=0 \). One checks \( u_x(0,0)=v_y(0,0)=1 \), \( u_y(0,0)=-v_x(0,0)=0 \), so CR holds at the origin and all partials exist there, yet \( f(z)/z = \bar z^2/z^2 \) has no limit as \( z\to0 \) (it equals \( e^{-4i\theta} \) along \( z=re^{i\theta} \)), so \( f \) is not complex differentiable at \( 0 \). Continuity of the partials near the point rules this pathology out by upgrading "partial derivatives exist" to genuine real (Fréchet) differentiability of \( (u,v):\mathbb{R}^2\to\mathbb{R}^2 \), via the standard mean-value-theorem lemma.

\( u,v \) real-valued. The decomposition \( f=u+iv \) with \( u,v:\Omega\to\mathbb{R} \) is what lets "\( f \)" be reinterpreted as a map \( \mathbb{R}^2\to\mathbb{R}^2 \); without insisting \( u,v \) are real (e.g. treating \( f \) itself and \( \bar f \) as the two "components") the linear-algebra argument identifying \( \mathbb{C} \)-linearity with the CR system breaks down.

Proof

Necessity. Suppose \( f'(z_0) = \lim_{h\to0} \dfrac{f(z_0+h)-f(z_0)}{h} \) exists, \( h\in\mathbb{C}\setminus\{0\} \).

This is the definition of complex differentiability at \( z_0 \): the difference quotient has a single limit as \( h\to 0 \) through arbitrary complex values, not just real or purely imaginary ones. A

Take \( h=t\in\mathbb{R} \), \( t\to 0 \): \( f'(z_0) = \lim_{t\to0}\dfrac{f(z_0+t)-f(z_0)}{t} = \lim_{t\to0}\dfrac{u(x_0+t,y_0)-u(x_0,y_0)}{t} + i\lim_{t\to0}\dfrac{v(x_0+t,y_0)-v(x_0,y_0)}{t}. \)

Restricting a convergent limit to a subset of directions gives the same value (subsequence/subnet principle for limits); splitting the complex quotient into real and imaginary parts is legitimate because \( t \) is real, so both fractions are genuine real difference quotients along the \( x \)-direction. This identifies both one-sided partials \( u_x(x_0,y_0),v_x(x_0,y_0) \) as existing, with \( f'(z_0) = u_x(x_0,y_0) + iv_x(x_0,y_0) \). B

Take \( h=is \), \( s\in\mathbb{R} \), \( s\to0 \): \( f'(z_0) = \lim_{s\to0}\dfrac{f(z_0+is)-f(z_0)}{is} = \dfrac{1}{i}\Big[\lim_{s\to0}\dfrac{u(x_0,y_0+s)-u(x_0,y_0)}{s} + i\lim_{s\to0}\dfrac{v(x_0,y_0+s)-v(x_0,y_0)}{s}\Big] = v_y(x_0,y_0) - iu_y(x_0,y_0). \)

Same restriction argument along the purely imaginary direction, then \( 1/i=-i \) is used to clear the denominator. This identifies \( u_y(x_0,y_0),v_y(x_0,y_0) \) as existing, with \( f'(z_0)=v_y(x_0,y_0)-iu_y(x_0,y_0) \). B

Equate the two expressions for \( f'(z_0) \): \( u_x(x_0,y_0)+iv_x(x_0,y_0) = v_y(x_0,y_0) - iu_y(x_0,y_0). \) Matching real and imaginary parts gives \( u_x=v_y \) and \( v_x=-u_y \) at \( (x_0,y_0) \).

A single well-defined complex number has a unique real part and a unique imaginary part, so two expressions for it must agree componentwise. This is exactly the Cauchy–Riemann system, completing necessity. A

Sufficiency, set-up. Assume \( u_x,u_y,v_x,v_y \) exist near \( (x_0,y_0) \), are continuous at \( (x_0,y_0) \), and satisfy CR there. By the differentiability lemma for functions of several real variables, continuity of all first partials at a point implies \( u \) and \( v \) are (Fréchet) differentiable at \( (x_0,y_0) \): writing \( \Delta x = x-x_0,\ \Delta y=y-y_0 \), \[ u(x,y) = u(x_0,y_0) + u_x\Delta x + u_y \Delta y + \varepsilon_1(\Delta x,\Delta y), \qquad v(x,y) = v(x_0,y_0) + v_x\Delta x + v_y\Delta y + \varepsilon_2(\Delta x,\Delta y), \] where \( \varepsilon_1,\varepsilon_2 = o(|\Delta x|+|\Delta y|) \) as \( (\Delta x,\Delta y)\to(0,0) \), all partials evaluated at \( (x_0,y_0) \).

This cites the standard multivariable-calculus lemma ("\( C^1 \) at a point \( \Rightarrow \) differentiable at that point"), itself proved via the one-variable Mean Value Theorem applied along each axis in turn plus continuity of the partials to control the cross term. It is the load-bearing analytic input that the point-counterexample in Hypotheses shows cannot be dropped. C

Combine into \( \Delta f := f(z_0+h)-f(z_0) \) for \( h=\Delta x+i\Delta y \): \[ \Delta f = (u_x\Delta x + u_y\Delta y) + i(v_x\Delta x + v_y\Delta y) + \varepsilon_1+i\varepsilon_2. \]

Direct substitution of the two real Taylor expansions from Step 5 into \( \Delta f = \Delta u + i\Delta v \). A

Apply CR (\( u_y=-v_x,\ v_y=u_x \)) to the bracketed linear part: \[ (u_x\Delta x + u_y\Delta y) + i(v_x\Delta x + v_y\Delta y) = u_x\Delta x - v_x\Delta y + i(v_x\Delta x + u_x\Delta y) = (u_x+iv_x)(\Delta x + i\Delta y) = (u_x+iv_x)\,h. \]

Pure algebraic substitution of the CR relations followed by regrouping into the form \( (\Delta x+i\Delta y)(u_x+iv_x) \), verified by expanding \( (u_x+iv_x)(\Delta x+i\Delta y)=u_x\Delta x - v_x\Delta y + i(u_x\Delta y+v_x\Delta x) \) and comparing term by term. This is the crux: CR is precisely the condition making the real-linear map \( (\Delta x,\Delta y)\mapsto(u_x\Delta x+u_y\Delta y,\ v_x\Delta x+v_y\Delta y) \) equal to multiplication by the single complex number \( u_x+iv_x \). B

Hence \( \Delta f = (u_x+iv_x)h + \varepsilon(h) \) with \( \varepsilon(h) := \varepsilon_1+i\varepsilon_2 = o(|h|) \), since \( |\Delta x|+|\Delta y| \le 2|h| \). Dividing by \( h\neq0 \): \[ \frac{f(z_0+h)-f(z_0)}{h} = u_x(x_0,y_0)+iv_x(x_0,y_0) + \frac{\varepsilon(h)}{h}, \qquad \left|\frac{\varepsilon(h)}{h}\right| \le \frac{|\varepsilon_1|+|\varepsilon_2|}{|h|} \to 0. \]

Uses \( |\Delta x|,|\Delta y|\le|h|\le|\Delta x|+|\Delta y| \) (equivalence of the max-norm/taxicab-norm and Euclidean norm on \( \mathbb{R}^2 \)) to convert the little-\( o \) statement in \( (\Delta x,\Delta y) \) into a little-\( o \) statement in \( h\in\mathbb{C} \), then the triangle inequality bounds the error term uniformly over the direction of \( h \), not just along axes. This is exactly the extra work needed beyond Steps 2–4: the limit must hold for \( h\to0 \) from every direction simultaneously. C

Therefore \( \displaystyle\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h} = u_x(x_0,y_0)+iv_x(x_0,y_0) \) exists, i.e. \( f \) is complex differentiable at \( z_0 \) with \( f'(z_0)=u_x(x_0,y_0)+iv_x(x_0,y_0) \).

By the definition of complex derivative (Step 1 reversed): the difference quotient converges to a fixed complex number independent of the direction of approach, as established in Step 8. This completes sufficiency. A

Result

f=u+iv \text{ is complex differentiable at } z_0 \iff u,v \text{ are (real-)differentiable at } (x_0,y_0) \text{ and } u_x=v_y,\ u_y=-v_x \text{ there.}

Reading. Writing a complex map as a pair of real functions of two real variables, complex differentiability at a point is equivalent to that pair being an ordinary differentiable map of \( \mathbb{R}^2\to\mathbb{R}^2 \) whose Jacobian matrix has the special form \( \begin{pmatrix} a & -b \\ b & a\end{pmatrix} \) — a scaling-and-rotation matrix — rather than a general \( 2\times2 \) matrix. The two scalar CR equations are exactly the two linear constraints cutting the four-dimensional space of Jacobians down to this two-dimensional (complex, \( a+ib \)) family.

Scope. The necessity half (Steps 1–4) requires only that \( f \) be complex differentiable at the single point \( z_0 \); no continuity of derivatives is needed. The sufficiency half (Steps 5–9) needs the partials to exist near \( z_0 \) and be continuous at \( z_0 \); this can be weakened (Looman–Menchoff: continuity of \( u,v \) plus CR holding a.e., plus partials existing, suffices) but that refinement is a separate, harder theorem and is not proved here. When CR holds at every point of an open set with continuous partials throughout, \( f \) is holomorphic on that set.

Corollaries & converses

If \( f=u+iv \) is holomorphic on a domain \( \Omega \) (so, by Goursat's theorem, \( f\in C^\infty(\Omega) \)), then \( u,v \) are \( C^2 \) and CR gives \( u_{xx}=v_{yx}=v_{xy}=-u_{yy} \), i.e. \( u \) (and likewise \( v \)) is harmonic: \( \Delta u = u_{xx}+u_{yy}=0 \). Real and imaginary parts of holomorphic functions are always harmonic conjugates.
If \( f'(z)=0 \) throughout a domain \( \Omega \), CR forces the Jacobian of \( (u,v) \) to vanish identically, so \( u,v \) are constant on each connected component of \( \Omega \); hence \( f \) is (locally) constant.
The converse of the necessity direction alone is false in the strong pointwise sense: existence of \( u_x,u_y,v_x,v_y \) at \( z_0 \) together with CR at \( z_0 \) does not imply \( f \) is complex differentiable at \( z_0 \) — see the Fails-without list. Continuity of the partials (or an equivalent regularity condition) is genuinely needed, so the theorem is a biconditional only under the stated extra hypothesis, not under CR alone.
An entire function with \( \bar f \) also holomorphic (equivalently \( f \) and \( \bar f \) both satisfy CR) must be constant, since CR for \( f \) gives \( u_x=v_y,u_y=-v_x \) while CR for \( \bar f = u-iv \) gives \( u_x=-v_y,u_y=v_x \), forcing \( u_x=u_y=v_x=v_y=0 \).

Fails without

Dropping continuity of the partials (keeping mere existence + CR at the point): \( f(z)=\bar z^2/z \) for \( z\neq0 \), \( f(0)=0 \), satisfies CR and has all partials existing at \( 0 \) (computed above), yet \( f(re^{i\theta})/re^{i\theta} = e^{-4i\theta} \) depends on \( \theta \), so \( \lim_{z\to0} f(z)/z \) does not exist: \( f \) is not complex differentiable at \( 0 \).
Dropping CR itself (real differentiability alone): \( f(z)=\bar z = x-iy \) has \( u=x,v=-y \), both real-differentiable everywhere with continuous (constant) partials, but \( u_x=1\neq -1=v_y \), so CR fails at every point; indeed \( \bar z \) is nowhere complex differentiable (\( \frac{f(z_0+h)-f(z_0)}{h}=\bar h/h \), which equals \( 1 \) for real \( h \) and \( -1 \) for purely imaginary \( h \)).
Restricting to a non-open domain (dropping "\( \Omega \) open near \( z_0 \)"): if \( f \) is only ever evaluated along the real axis, \( u_y,v_y \) are not even defined, so "CR holds" is vacuous or ill-posed and cannot certify complex differentiability of any genuine extension to a neighbourhood of \( z_0 \) in \( \mathbb{C} \).

Common errors

Treating "\( u,v \) satisfy CR at \( z_0 \)" alone as sufficient for \( f'(z_0) \) to exist, forgetting the continuity-of-partials hypothesis (or full real-differentiability) needed for the converse direction — the classic trap illustrated by \( \bar z^2/z \).
Getting the sign wrong: writing \( u_y=v_x \) instead of \( u_y=-v_x \); this is usually caught by re-deriving from \( f'(z_0)=u_x+iv_x=v_y-iu_y \) rather than memorising the formula.
Applying CR to \( f(z)=\bar z \), \( |z|^2 \), \( \operatorname{Re}(z) \), or \( \operatorname{Im}(z) \) and concluding they are holomorphic because they are smooth as maps \( \mathbb{R}^2\to\mathbb{R}^2 \) — smoothness is not the issue, satisfying the linear constraint CR imposes on the Jacobian is.
Forgetting that CR is stated with respect to the standard identification \( z=x+iy \); using CR verbatim after substituting polar coordinates without switching to the polar form \( u_r = \tfrac{1}{r}v_\theta,\ v_r=-\tfrac1r u_\theta \) (which is a distinct, though equivalent, pair of equations).
Concluding \( u \) is harmonic directly from CR without first knowing \( f \) is holomorphic on an open set (so that Goursat guarantees \( u,v\in C^2 \)); CR alone at isolated points, or without the regularity needed to interchange mixed partials (Clairaut/Schwarz), does not license \( u_{xy}=u_{yx} \).

Discussion

Historically the equations bear the names of Augustin-Louis Cauchy, who used them (1814 onward) in the context of integral formulas for what would become complex analysis, and Bernhard Riemann, whose 1851 dissertation placed them at the foundation of a geometric theory of functions of a complex variable, treating holomorphy as a local conformality condition rather than a formula-based notion. The equations had, in fact, appeared earlier still in d'Alembert's work on fluid dynamics (1752) — they are precisely the irrotational, incompressible flow condition for the velocity field \( (u,-v) \), which is one reason harmonic conjugate pairs \( (u,v) \) show up throughout 2D potential theory, electrostatics, and fluid mechanics.

Conceptually, CR is the statement that the real-linear approximation to \( f \) at \( z_0 \) (its Jacobian, an element of \( \mathrm{Mat}_2(\mathbb{R}) \)) actually lies in the subalgebra of matrices of the form \( \begin{pmatrix}a&-b\\b&a\end{pmatrix} \), which is isomorphic as an \( \mathbb{R} \)-algebra to \( \mathbb{C} \) itself (via \( a+ib \leftrightarrow \) the matrix). So "\( f \) is complex differentiable" literally means "the best real-linear approximation to \( f \) is multiplication by a complex number," which is a strictly more rigid condition than being an arbitrary real-linear map (that four-real-parameter freedom being cut to two real parameters, i.e. one complex parameter, by the two CR constraints). This rigidity is what ultimately produces the entire battery of strong theorems in complex analysis (analyticity, Cauchy integral formula, identity theorem) that have no counterpart for merely differentiable maps \( \mathbb{R}^2\to\mathbb{R}^2 \).

A cleaner formal packaging uses the Wirtinger derivatives \( \partial_{\bar z} = \tfrac12(\partial_x+i\partial_y) \), \( \partial_z=\tfrac12(\partial_x-i\partial_y) \), formally treating \( z,\bar z \) as independent variables. A short computation shows the CR system is equivalent to the single complex equation \( \partial_{\bar z} f = 0 \), and when this holds, \( f'(z_0)=\partial_z f(z_0) \). This reformulation both explains the mnemonic "\( f \) holomorphic \( \iff \) \( f \) does not depend on \( \bar z \)" and generalises immediately to CR-type systems in several complex variables.

Common misconception: that "\( u,v \) satisfy CR everywhere" is by itself synonymous with "\( f \) is holomorphic." The precise statement (this theorem, sufficiency direction) needs continuity of the partial derivatives, or some substitute regularity hypothesis; the sharpest classical repair, the Looman–Menchoff theorem, shows that mere continuity of \( f \) together with CR holding at every point (partials assumed to exist, not assumed continuous) is enough to force holomorphy — a genuinely deeper real-analytic fact whose proof is well beyond the differentiability-lemma argument used above, and is not established by this theorem.

Worked examples

Show \( f(z)=z^2 \) is holomorphic on \( \mathbb{C} \) via CR, and find \( f' \).

Write \( z^2=(x+iy)^2 = (x^2-y^2)+i(2xy) \), so \( u(x,y)=x^2-y^2,\ v(x,y)=2xy \). A

\( u_x=2x,\ u_y=-2y,\ v_x=2y,\ v_y=2x. \) These are polynomials in \( x,y \), hence continuous everywhere.

Direct differentiation; polynomials are \( C^\infty \), satisfying the continuity hypothesis of the theorem at every point of \( \mathbb{C} \). A

Check CR: \( u_x=2x=v_y \) ✓, and \( u_y=-2y=-(2y)=-v_x \) ✓, at every \( (x,y) \).

Direct substitution of Step 2 into the two CR identities. A

By the theorem (sufficiency direction, applied at every point of the open set \( \mathbb{C} \)), \( f \) is complex differentiable everywhere, with \( f'(z) = u_x+iv_x = 2x+i(2y) = 2(x+iy) = 2z. \)

Direct application of the sufficiency conclusion \( f'(z_0)=u_x(x_0,y_0)+iv_x(x_0,y_0) \), then resubstituting \( x+iy=z \). B

z^2 \text{ is entire, with } (z^2)'=2z, \text{ confirmed independently of the power rule by direct CR verification.}

Determine every point where \( f(z) = x^2+y^2 + i(2xy) \) (i.e. \( u=x^2+y^2,\ v=2xy \)) is complex differentiable.

Set-up: identify \( u,v \) as given, both polynomials hence \( C^\infty \) (continuous partials everywhere), so only CR itself needs checking pointwise. A

Compute \( u_x=2x,\ u_y=2y,\ v_x=2y,\ v_y=2x. \)

Direct differentiation of the given polynomials. A

CR equation 1: \( u_x=v_y \) becomes \( 2x=2x \), always true. CR equation 2: \( u_y=-v_x \) becomes \( 2y=-2y \), i.e. \( 4y=0 \), i.e. \( y=0 \).

Substituting Step 2 into each CR equation and solving; the first is an identity, the second isolates the locus \( y=0 \) (the real axis). B

Since the partials are continuous everywhere (Step 1), the theorem's sufficiency direction applies at every point of the locus \( \{y=0\} \): \( f \) is complex differentiable exactly at points \( z_0=x_0 \) with \( x_0\in\mathbb{R} \), and there \( f'(x_0) = u_x+iv_x = 2x_0 + i(0) = 2x_0. \) At any point with \( y\neq0 \), CR fails, so by the necessity direction (contrapositive) \( f \) is not complex differentiable there.

Necessity direction (Steps 1–4 of the Proof) used in contrapositive form off the real axis; sufficiency direction used on the real axis, where continuity of partials was already verified in Step 1 here. Since \( f \) is complex differentiable only on the non-open set \( \mathbb{R}\subset\mathbb{C} \) and not on any open neighbourhood of any point, \( f \) is nowhere holomorphic (holomorphy requires differentiability on an open set). C

f \text{ is complex differentiable only on } \mathbb{R}\subset\mathbb{C}, \text{ with } f'(x_0)=2x_0 \text{ there; } f \text{ is holomorphic nowhere.}

Problems

Show that \( f(z)=\overline{z}\,e^{z} \) is nowhere complex differentiable.
Solution
Write \( e^{x+iy}=e^x\cos y + ie^x\sin y \), so \( \bar z e^z = (x-iy)(e^x\cos y+ie^x\sin y) \). Expanding: \( u = e^x(x\cos y + y\sin y) \) [real part], \( v = e^x(x\sin y - y\cos y) \) [imaginary part]. Rather than expand fully, use the corollary-style shortcut: \( g(z):=\bar z \) has Jacobian \( \begin{pmatrix}1&0\\0&-1\end{pmatrix} \) (satisfies CR nowhere, as shown in Fails-without), while \( h(z):=e^z \) is entire and \( h(z)\neq0 \) everywhere. If \( f=g\cdot h \) were complex differentiable at some \( z_0 \), then since \( h(z_0)\neq0 \) and \( h \) is holomorphic near \( z_0 \), \( g = f/h \) would be a quotient of complex-differentiable functions at \( z_0 \) (using the quotient rule, valid once both factors are known complex differentiable at \( z_0 \)) hence complex differentiable at \( z_0 \) — contradicting that \( \bar z \) is nowhere complex differentiable. So \( f \) is nowhere complex differentiable. (Equivalently, compute \( u,v \) explicitly and check CR fails at every point — a longer but elementary route.)
Let \( f(z) = x^3 - 3xy^2 + i(3x^2y - y^3) \). Verify CR holds everywhere and find \( f'(z) \) in terms of \( z \).
Solution
\( u=x^3-3xy^2,\ v=3x^2y-y^3 \). Then \( u_x=3x^2-3y^2,\ u_y=-6xy,\ v_x=6xy,\ v_y=3x^2-3y^2 \). CR: \( u_x=v_y \) gives \( 3x^2-3y^2=3x^2-3y^2 \) ✓; \( u_y=-v_x \) gives \( -6xy=-6xy \) ✓. All partials are polynomials, hence continuous everywhere, so by the theorem \( f \) is entire with \( f'(z)=u_x+iv_x = (3x^2-3y^2)+i(6xy) = 3(x^2-y^2+2ixy)=3(x+iy)^2=3z^2 \). (Indeed \( f(z)=z^3 \), since \( z^3=(x+iy)^3=x^3+3x^2(iy)+3x(iy)^2+(iy)^3=x^3-3xy^2+i(3x^2y-y^3) \).)
Suppose \( f=u+iv \) is entire and \( u(x,y) = x^2-y^2 \) for all \( (x,y) \). Using CR, find the most general possible \( v \), and hence \( f \), given \( f(0)=i \).
Solution
\( u_x=2x=v_y \Rightarrow v = 2xy + \varphi(x) \) for some function \( \varphi \) (integrating in \( y \) with \( x \) held fixed). Also \( u_y=-2y \) must equal \( -v_x \), i.e. \( v_x = 2y \). But from \( v=2xy+\varphi(x) \), \( v_x = 2y+\varphi'(x) \). Matching: \( 2y+\varphi'(x)=2y \Rightarrow \varphi'(x)=0 \Rightarrow \varphi(x)=c \) constant. So \( v=2xy+c \), giving \( f(z) = (x^2-y^2)+i(2xy+c) = z^2+ic \) (using \( x^2-y^2+2ixy=z^2 \)). Impose \( f(0)=i \): \( f(0)=0+ic=ic=i \Rightarrow c=1 \). So \( f(z)=z^2+i \). (This is the harmonic-conjugate construction: CR determines \( v \) up to an additive real constant once \( u \) is fixed and \( \Omega \) is connected.)
Determine whether there exists an entire function \( f \) with \( \operatorname{Re}(f(z)) = xy \) for all \( z \), by attempting the harmonic-conjugate construction and checking consistency.
Solution
First check \( u=xy \) is harmonic (a necessary condition, by the Corollaries, for \( u \) to be the real part of a holomorphic function): \( u_{xx}=0,\ u_{yy}=0 \), so \( \Delta u=0 \) ✓, harmonicity holds, so no contradiction yet. Proceed: \( u_x=y=v_y \Rightarrow v=\tfrac12y^2+\varphi(x) \). Need \( u_y=x=-v_x=-\varphi'(x) \Rightarrow \varphi'(x)=-x \Rightarrow \varphi(x) = -\tfrac12x^2+c \). So \( v = \tfrac12y^2-\tfrac12x^2+c = -\tfrac12(x^2-y^2)+c \). Both determining equations were consistent (no contradiction forced \( \varphi'\) to be simultaneously two different non-constant things), so such \( f \) exists: \( f(z) = xy + i\big(-\tfrac12(x^2-y^2)+c\big) \). Checking against \( z^2=(x^2-y^2)+i(2xy) \): note \( -\tfrac{i}{2}z^2 = -\tfrac i2(x^2-y^2) + xy \), matching \( f \) exactly (with \( c=0 \)). So \( f(z) = -\tfrac{i}{2}z^2 + ic \) works for any real \( c \); in particular such an entire \( f \) exists (uniquely up to an additive purely imaginary constant, by the Corollaries).
Let \( f(z) = \sqrt{|xy|} \) (real-valued, so \( v\equiv0 \)). Show CR holds at \( z_0=0 \) but \( f \) is not complex differentiable there, and identify precisely which hypothesis of the theorem fails.
Solution
With \( v\equiv 0 \), \( v_x=v_y=0 \) everywhere. Compute \( u_x(0,0) = \lim_{t\to0}\frac{\sqrt{|t\cdot0|}-0}{t} = \lim_{t\to0}\frac{0}{t}=0 \), and symmetrically \( u_y(0,0)=0 \). So CR at the origin reads \( u_x=v_y \): \( 0=0 \) ✓, and \( u_y=-v_x \): \( 0=-0 \) ✓ — CR holds at \( (0,0) \). Now test complex differentiability directly along \( z=t(1+i) \), \( t\in\mathbb{R}\to0 \): \( f(t(1+i)) = \sqrt{|t\cdot t|} = |t| \), so \( \frac{f(z)-f(0)}{z} = \frac{|t|}{t(1+i)} \), whose modulus is \( \frac{1}{\sqrt2} \) for all \( t\neq0 \) but whose sign flips with the sign of \( t \) (since \( |t|/t = \pm1 \)); the limit as \( t\to0^+ \) and \( t\to0^- \) disagree, so \( \lim_{z\to0} f(z)/z \) does not exist and \( f \) is not complex differentiable at \( 0 \). The failing hypothesis is continuity of the partials at \( (0,0) \): one can check \( u_x(x,y) \) does not tend to \( 0 \) as \( (x,y)\to(0,0) \) along generic paths (e.g. along \( y=x \), \( u=\sqrt{x^2}=|x| \), which is not differentiable in \( x \) at \( x=0\) once \( y\ne0\) is allowed to vary, so \( u_x \) fails to be continuous at the origin), so the sufficiency direction's continuity hypothesis is not met, and indeed its conclusion fails — consistent with, and a further illustration of, the theorem.