Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 7

Evaluate the derivative along real and imaginary increments. For a holomorphic function, both give the same value:Equating real and imaginary parts yields the Cauchy-Riemann equations

Differentiate the Cauchy-Riemann equations and use equality of mixed partial derivatives, which holds because holomorphic functions are smooth:Thus both parts are harmonic functions.

The Cauchy-Riemann equations give $\nabla v=(-u_y,u_x)$. ThereforeThe gradient is normal to a regular level set, so these curves intersect orthogonally wherever their gradients are nonzero. At a stationary point, where both gradients vanish, a regular curve tangent need not exist; the algebraic identity remains valid.

The rotation $\nabla v=(-u_y,u_x)$ from the Cauchy-Riemann equations is invertible. HenceThus their stationary points coincide, including degenerate ones.

For a harmonic function, put $a=u_{xx}$ and $b=u_{xy}=u_{yx}$. Since $u_{yy}=-u_{xx}$, its Hessian matrix isA nonzero determinant is therefore negative, and the two real Hessian eigenvalues are $\pm \sqrt{a^2+b^2}$. At a stationary point the quadratic term of the Taylor expansion has both signs. Consequently every such nondegenerate stationary point is a saddle point. This is the nondegenerate stationary points of a planar harmonic function criterion. The same argument applies to $v$, whose stationary points coincide with those of $u$. A zero Hessian determinant is outside this second-order classification.

Assume the image of the domain of $f$ lies in the domain where $g$ is holomorphic. The complex chain rule givesso the composition is holomorphic, including at points where $f^{\prime }=0$. Its real part $S(x,y)=s(u(x,y),v(x,y))$ is therefore a harmonic function.

A direct verification uses the composition of harmonic conjugate pairs. The multivariable chain rule givesThe Cauchy-Riemann equations imply $|\nabla u{|}^2=|\nabla v{|}^2$; parts (i) and (ii) remove the last three terms. Since $s$ is harmonic in its own variables,