Proof that the real part of a holomorphic function is harmonic

Builds on Holomorphic function Holomorphic functions are infinitely differentiable Cauchy–Riemann equations Schwarz's theorem for C^2 functions C^k function characterised by continuous derivatives Harmonic function Real and imaginary parts

Proof

We differentiate the Cauchy–Riemann equations once more and use the symmetry of mixed partial derivatives.

Since $f$ is holomorphic on $U$ , holomorphic functions are infinitely differentiable. So, as a function of $(x, y)$ , $f$ has continuous partial derivatives of all orders on $U$ .
The partial derivatives of $f$ are those of $u$ plus $i$ times those of $v$ , by the definition of real and imaginary parts. So all partial derivatives of $u$ and $v$ exist and are continuous on $U$ .
In particular, $u$ and $v$ are of class $C^2$ on $U$ . This is the first claim.
Since $f$ is complex differentiable at each point of $U$ , the Cauchy–Riemann equations hold at each point of $U$ : $u_x = v_y$ and $u_y = -v_x$ on $U$ .
These are equalities of functions on the open set $U$ . So we may differentiate both sides. Differentiating $u_x = v_y$ in $x$ gives $u_{xx} = (v_y)_x$ .
Differentiating $u_y = -v_x$ in $y$ gives $u_{yy} = -(v_x)_y$ .
Since $v$ is of class $C^2$ , Schwarz's theorem gives $(v_y)_x = (v_x)_y$ on $U$ .
Adding steps 5 and 6 and using step 7 gives $u_{xx} + u_{yy} = (v_y)_x - (v_x)_y = 0$ on $U$ .
Differentiating $u_y = -v_x$ in $x$ gives $(u_y)_x = -v_{xx}$ . Differentiating $u_x = v_y$ in $y$ gives $(u_x)_y = v_{yy}$ .
Schwarz's theorem for $u$ gives $(u_y)_x = (u_x)_y$ . So $v_{xx} + v_{yy} = -(u_y)_x + (u_x)_y = 0$ on $U$ .
By steps 3, 8 and 10, $u$ and $v$ are of class $C^2$ with zero Laplacian. So they are harmonic on $U$ .

Idea

The Cauchy–Riemann equations tie the first derivatives of $u$ to those of $v$ . Differentiating once more, the two second derivatives $u_{xx}$ and $u_{yy}$ become the same mixed derivative of $v$ with opposite signs. The real work is hidden in step 1: we need to know that $f$ has second derivatives at all.

Reading path

To understand this page from the ground up, read these 455 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the Laplacian as a sum of second partial derivatives, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule in several variables, Proof of the chain rule with an auxiliary difference quotient, Proof of the reverse triangle inequality, Proof of the derivative as a little-o approximation, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that holomorphic functions are infinitely differentiable, Proof that Lipschitz functions are uniformly continuous, Proof that the integral of a continuous function is an antiderivative, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof of the limit of a product of sequences, Proof of Cauchy's integral formula by a primitive of the difference quotient, Proof of the linearity of the derivative, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the derivative of a continuous linear map, Proof of the Cauchy–Riemann equations, Proof of the existence of square roots of nonnegative reals, Proof of the chain rule for a composition with a curve, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof that a continuous image of a compact set is compact, Proof that compact subsets of a Hausdorff space are closed, Proof that the composition of continuous maps is continuous, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of the description of the open sets of a metric space, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof of Schwarz's theorem for C^2 functions by the mean value theorem, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof of the comparison test, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof that every spanning set contains a basis, by Zorn's lemma, Proof of the estimation lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of integration by substitution, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that the derivative gives the directional derivatives, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof that differentiable functions are continuous, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the fundamental theorem of contour integration, Proof of Goursat's theorem by repeated subdivision, Proof of the Archimedean property in an Archimedean ordered field, Proof that a continuous real function on a compact set attains its bounds, Proof of the derivative of a map into a product, Proof that a composition of continuous functions is continuous, Proof of Cantor's intersection theorem, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that holomorphic functions are analytic, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that absolutely convergent series in a Banach space converge, Proof of the symmetry of second derivatives by a second difference, Proof that open balls are open, Proof that power series are holomorphic in the disc of convergence, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that C^k functions are characterised by continuous derivatives, Proof that continuous partial derivatives imply differentiability, Proof that a set is open if and only if its complement is closed, Proof of the fundamental theorem of calculus, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that sums and products of continuous functions are continuous, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the mean value inequality, Proof of the constant multiple rule.

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Proof of the epsilon-delta form of continuity proof
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Proof that a function is bijective iff it has a two-sided inverse proof
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Proof that Lipschitz functions are uniformly continuous proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
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Proof that the composition of continuous maps is continuous proof
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Proof of the limit of a sum of sequences proof
Compactness via open covers statement
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Proof of the uniqueness of limits proof
Proof of the squeeze theorem for sequences proof
Proof that metric spaces are Hausdorff proof
Proof of continuity via open sets in metric spaces proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof that limits preserve non-strict inequalities proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
Proof that subsequences of a convergent sequence converge to the same limit proof
The reals are Archimedean statement
Proof of the monotone convergence theorem for sequences proof
Complex differentiability definition
Proof that compact subsets of a Hausdorff space are closed proof
Reverse triangle inequality statement
Proof that a complex number times its conjugate is the square of its modulus proof
Proof of Cantor's intersection theorem proof
Proof that a continuous image of a compact set is compact proof
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Epsilon-delta continuity in metric spaces statement
Lebesgue number lemma statement
Proof that compactness and sequential compactness agree in metric spaces proof
Differentiable functions are continuous statement
Proof of the sequence characterisation of uniform continuity proof
Proof of the triangle inequality for norms proof
Limit of a sum of sequences statement
Uniqueness of limits statement
Linearity of the integral statement
Limits preserve non-strict inequalities statement
Proof of the monotonicity of the integral proof
Squeeze theorem statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Convergent sequences are bounded statement
Subsequences of a convergent sequence converge to the same limit statement
Proof that Darboux and Riemann integrals agree proof
Convergent sequences are Cauchy statement
Monotone convergence theorem for sequences statement
Holomorphic function definition
A complex number times its conjugate statement
Compact subsets of a Hausdorff space are closed statement
Cantor's intersection theorem statement
Proof of the Cauchy–Riemann equations proof
Continuous image of a compact set is compact statement
Compact subsets of a metric space are bounded statement
Continuity via open sets statement
Compactness and sequential compactness agree in metric spaces statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon statement
Triangle inequality for norms statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Cauchy–Riemann equations statement
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof that Lipschitz maps are uniformly continuous proof
Proof that absolutely convergent series in a Banach space converge proof
Proof of the sequential characterization of limits proof
Proof of the Cauchy–Schwarz inequality proof
Proof that continuity is equivalent to sequential continuity proof
Geometric series example
Proof of Riemann's criterion for integrability proof
Limit of a product of sequences statement
Decreasing sequence bounded below converges statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Absolutely convergent series in a Banach space converge statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Sequential characterization of limits statement
Proof of the ratio test proof
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Comparison test statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Proof of the derivative of a map into a product proof
Product of continuous functions is continuous statement
Chain rule statement
Ratio test statement
Proof of the linearity of the derivative proof
Additivity of the integral over intervals statement
Proof of the Heine–Borel theorem by sequential compactness proof
Operator norm definition
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Linearity of differentiation statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Proof that power series are holomorphic in the disc of convergence proof
Linearity of the derivative statement
Heine–Borel theorem statement
Continuous functions on a closed bounded interval are uniformly continuous statement
Boundedness theorem statement
Proof of the constant multiple rule proof
Proof of the first fundamental theorem of calculus proof
Power series are holomorphic in the disc of convergence statement
Derivative of a map into R^n by components statement
Constant multiple rule statement
Proof that a continuous real function on a compact set attains its bounds proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Proof of Fermat's theorem on stationary points proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Proof that continuous functions are Riemann integrable proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Equivalence of norms in finite dimension statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof of the estimation lemma proof
Proof of the mean value theorem for integrals proof
Estimation lemma statement
Linear maps on finite-dimensional spaces are continuous statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof of the chain rule in several variables proof
Proof that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
Proof that the integral of a continuous function is an antiderivative proof
Proof of the second fundamental theorem of calculus proof
The integral of a continuous function is an antiderivative statement
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Chain rule for a composition with a curve statement
Proof that C^k functions are characterised by continuous derivatives proof
Proof of integration by substitution proof
C^k function characterised by continuous derivatives statement
Integration by substitution statement
Fundamental theorem of calculus statement
Proof of the mean value inequality proof
Proof of the fundamental theorem of contour integration proof
Proof of Schwarz's theorem for C^2 functions by the mean value theorem proof
Vector field definition
Mean value inequality statement
Fundamental theorem of contour integration statement
Divergence definition
Proof of the symmetry of second derivatives by a second difference proof
Laplacian definition
Proof of Goursat's theorem by repeated subdivision proof
Symmetry of second derivatives statement
Goursat's theorem statement
Schwarz's theorem for C^2 functions statement
Proof of the Laplacian as a sum of second partial derivatives proof
Proof of Cauchy's integral formula by a primitive of the difference quotient proof
Laplacian as a sum of second partial derivatives statement
Cauchy's integral formula statement
Harmonic function definition
Proof that holomorphic functions are analytic proof
Holomorphic functions are analytic statement
Proof that holomorphic functions are infinitely differentiable proof
Holomorphic functions are infinitely differentiable statement
Proof that the real part of a holomorphic function is harmonic proof

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59 pages build on this one.

Tag	`XY7D3X10QR`
URI	`m2://XY7D3X10QR`
Kind	proof
Writing	W0 stub
MSC	31A05
Proves	The real part of a holomorphic function is harmonic
Uses	Holomorphic function, Holomorphic functions are infinitely differentiable, Cauchy–Riemann equations, Schwarz's theorem for C^2 functions, C^k function characterised by continuous derivatives, Harmonic function, Real and imaginary parts
Changed	4 November 2026

Proof that the real part of a holomorphic function is harmonic

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