The mixed boundary value problem can be reduced to an RH problem as follows. Let the boundary values of the multi-valued analytic function

*F* be given by

$F=\psi +\mathrm{i}\phi .$

(20)

Although, the function

$F(z)$ is in general multi-valued, its derivative

${F}^{\prime}$ is a single-valued analytic function on

*G*. The boundary values of the function

${F}^{\prime}(z)$ are given by

$\dot{\eta}{F}^{\prime}={\psi}^{\prime}+\mathrm{i}{\phi}^{\prime}.$

(21)

For the Dirichlet conditions,

*i.e.*,

$t\in {J}_{j}$ and

$j\in {S}_{\mathbf{d}}$, the functions

${\psi}_{j}$ are equal to the known functions

${\varphi}_{j}(t)$ (see (6b)). Thus, the function

$F(z)$ satisfies the RH problem

$Re\left[F({\eta}_{j}(t))\right]={\varphi}_{j}(t),\phantom{\rule{1em}{0ex}}t\in {J}_{j},j\in {S}_{\mathbf{d}}.$

(22)

The Neumann conditions can also be reduced to an RH problem by using the Cauchy-Riemann equations and integrating along the boundaries

${\mathrm{\Gamma}}_{j}$,

$j\in {S}_{\mathbf{n}}$. Let

$\mathbf{T}(\zeta )$ be the unit tangent vector and

$\mathbf{n}(\zeta )$ be the unit external normal vector to Γ at

$\zeta \in \mathrm{\Gamma}$. Let also

$\nu (\zeta )$ be the angle between the normal vector

$\mathbf{n}(\zeta )$ and the positive real axis,

*i.e.*,

$\mathbf{n}(\zeta )={e}^{\mathrm{i}\nu (\zeta )}$. Then

${e}^{\mathrm{i}\nu (\eta (t))}=-\mathrm{i}\mathbf{T}(\eta (t))=-\mathrm{i}\frac{\dot{\eta}(t)}{|\dot{\eta}(t)|}.$

Thus,

$\frac{\partial u}{\partial \mathbf{n}}=\mathrm{\nabla}u\cdot \mathbf{n}=cos\nu \frac{\partial u}{\partial x}+sin\nu \frac{\partial u}{\partial y}=Re\left[{e}^{\mathrm{i}\nu}(\frac{\partial u}{\partial x}-\mathrm{i}\frac{\partial u}{\partial y})\right].$

(23)

Since

$u(z)=ReF(z)$, then by the Cauchy-Riemann equations, we have

${F}^{\prime}(z)=\frac{\partial u(z)}{\partial x}-\mathrm{i}\frac{\partial u(z)}{\partial y}.$

Thus, the function

${F}^{\prime}(z)$ satisfies the RH problem

$Re[-\mathrm{i}{\dot{\eta}}_{j}(t){F}^{\prime}]=|{\dot{\eta}}_{j}(t)|\frac{\partial u}{\partial \mathbf{n}},\phantom{\rule{1em}{0ex}}t\in {J}_{j},j\in {S}_{\mathbf{n}}.$

(24)

If we define the real piecewise constant function

$\theta (t)=\{\begin{array}{cc}0,\hfill & t\in {J}_{j},j\in {S}_{\mathbf{d}},\hfill \\ \pi /2,\hfill & t\in {J}_{j},j\in {S}_{\mathbf{n}},\hfill \end{array}$

(25)

the boundary values of the function

$F(z)$ satisfy on the boundary Γ the condition

$Re\left[{e}^{-\mathrm{i}\theta (t)}F(\eta (t))\right]=\stackrel{\u02c6}{\varphi}(t),$

(26)

where

$\stackrel{\u02c6}{\varphi}(t)=\{\begin{array}{cc}{\varphi}_{j}(t),\hfill & t\in {J}_{j},j\in {S}_{\mathbf{d}},\hfill \\ {\phi}_{j}(t),\hfill & t\in {J}_{j},j\in {S}_{\mathbf{n}},\hfill \end{array}$

(27)

is known and

${\phi}_{j}^{\prime}(t)=Re[-\mathrm{i}{\dot{\eta}}_{j}(t){F}^{\prime}({\eta}_{j}(t))]={\varphi}_{j}(t)|{\dot{\eta}}_{j}(t)|,\phantom{\rule{1em}{0ex}}t\in {J}_{j},j\in {S}_{\mathbf{n}}.$

(28)

The functions

${\varphi}_{j}(t)$ for

$j\in {S}_{\mathbf{d}}\cup {S}_{\mathbf{n}}$ are given by (6b) and (6c). The functions

${\phi}_{j}(t)$ can be then computed for

$t\in {J}_{j}$ and

$j\in {S}_{\mathbf{n}}$ by integrating the functions

${\phi}_{j}^{\prime}(t)$. Then it follows from (7), (26) and (27) that the function

$f(z)$ is a solution of the RH problem

$Re\left[{e}^{-\mathrm{i}\theta (t)}f(\eta (t))\right]=\stackrel{\u02c6}{\varphi}(t)+\sum _{k=1}^{m}{a}_{k}Re[{e}^{-\mathrm{i}\theta (t)}log(\eta (t)-{z}_{k})],$

(29)

or briefly,

$Re\left[{e}^{-\mathrm{i}\theta (t)}f(\eta (t))\right]=\stackrel{\u02c6}{\varphi}(t)+\sum _{k\in {S}_{\mathbf{n}}}{a}_{k}{\gamma}^{[k]}(t)+\sum _{k\in {S}_{\mathbf{d}}}{a}_{k}{\gamma}^{[k]}(t),$

(30)

where

${\gamma}^{[k]}(t)=Re[{e}^{-\mathrm{i}\theta (t)}log(\eta (t)-{z}_{k})],$

(31)

for

$k=1,2,\dots ,m$. In view of (8) and (28), the real constants

${a}_{k}$ are known for

$k\in {S}_{\mathbf{n}}$ and are given by

${a}_{k}=\frac{1}{2\pi}{\int}_{{J}_{k}}{\varphi}_{k}(t)|{\dot{\eta}}_{k}(t)|\phantom{\rule{0.2em}{0ex}}dt,\phantom{\rule{1em}{0ex}}k\in {S}_{\mathbf{n}}.$

(32)

However, for

$k\in {S}_{\mathbf{d}}$, the real constants

${a}_{k}$ are unknown. Thus, the boundary condition (29) can be written as

$Re\left[{e}^{-\mathrm{i}\theta (t)}f(\eta (t))\right]=\stackrel{\u02c6}{\psi}(t)+\sum _{k\in {S}_{\mathbf{d}}}{a}_{k}{\gamma}^{[k]}(t),\phantom{\rule{1em}{0ex}}t\in J,$

(33)

where the function

$\stackrel{\u02c6}{\psi}(t)$ is known and is given by

$\stackrel{\u02c6}{\psi}(t)=\{\begin{array}{cc}{\varphi}_{j}(t)+{\sum}_{k\in {S}_{\mathbf{n}}}{a}_{k}{\gamma}_{j}^{[k]}(t),\hfill & t\in {J}_{j},j\in {S}_{\mathbf{d}},\hfill \\ {\phi}_{j}(t)+{\sum}_{k\in {S}_{\mathbf{n}}}{a}_{k}{\gamma}_{j}^{[k]}(t),\hfill & t\in {J}_{j},j\in {S}_{\mathbf{n}}.\hfill \end{array}$

(34)

Obviously, the functions ${\stackrel{\u02c6}{\psi}}_{j}(t)$ are known explicitly for $t\in {J}_{j}$ with $j\in {S}_{\mathbf{d}}$. However, for $t\in {J}_{j}$ with $j\in {S}_{\mathbf{n}}$, it is required to integrate ${\phi}_{j}^{\prime}(t)$ to obtain ${\phi}_{j}(t)$.

The functions

${\phi}_{j}(t)$ are not necessary 2

*π*-periodic. In order to keep dealing with periodic functions numerically, we do not compute

${\phi}_{j}(t)$ directly by integrating the functions

${\phi}_{j}^{\prime}(t)$. Instead, we integrate the functions

${\stackrel{\u02c6}{\psi}}_{j}^{\prime}(t)={\varphi}_{j}(t)|{\dot{\eta}}_{j}(t)|+\sum _{k\in {S}_{\mathbf{n}}}{a}_{k}\frac{d}{dt}{\gamma}_{j}^{[k]}(t).$

According to the definitions of the constants

${a}_{k}$ and the functions

${\gamma}^{[k]}$, we have

${\int}_{0}^{2\pi}{\stackrel{\u02c6}{\psi}}_{j}^{\prime}(t)\phantom{\rule{0.2em}{0ex}}dt=0,$

which implies that the functions

${\stackrel{\u02c6}{\psi}}_{j}(t)={\phi}_{j}(t)+{\sum}_{k\in {S}_{\mathbf{n}}}{a}_{k}{\gamma}_{j}^{[k]}(t)$ are always 2

*π*-periodic. By using the Fourier series for

$t\in {J}_{j}$ with

$j\in {S}_{\mathbf{n}}$, the functions

${\stackrel{\u02c6}{\psi}}_{j}^{\prime}(t)$ can be written as

${\stackrel{\u02c6}{\psi}}_{j}^{\prime}(t)=\sum _{i=1}^{\mathrm{\infty}}{a}_{i}^{[j]}cosit+\sum _{i=1}^{\mathrm{\infty}}{b}_{i}^{[j]}sinit.$

(35)

Then the functions

${\stackrel{\u02c6}{\psi}}_{j}(t)$ are given for

$t\in {J}_{j}$ with

$j\in {S}_{\mathbf{n}}$ by

${\stackrel{\u02c6}{\psi}}_{j}(t)={\tilde{\psi}}_{j}(t)+{c}_{j},$

(36)

where

${c}_{j}$ are undetermined real constants and the functions

${\tilde{\psi}}_{j}(t)$ are given by

${\tilde{\psi}}_{j}(t)=\sum _{i=1}^{\mathrm{\infty}}\frac{{a}_{i}^{[j]}}{i}sinit-\sum _{i=1}^{\mathrm{\infty}}\frac{{b}_{i}^{[j]}}{i}cosit,\phantom{\rule{1em}{0ex}}t\in {J}_{j},j\in {S}_{\mathbf{n}}.$

(37)

Hence, the boundary condition (33) can then be written as

$Re\left[{e}^{-\mathrm{i}\theta (t)}f(\eta (t))\right]=\stackrel{\u02c6}{\gamma}(t)+\tilde{h}(t)+\sum _{k\in {S}_{\mathbf{d}}}{a}_{k}{\gamma}^{[k]}(t),\phantom{\rule{1em}{0ex}}t\in J,$

(38)

where

$\tilde{h}(t)$ is the real piecewise constant function

$\tilde{h}(t)=\{\begin{array}{cc}0,\hfill & t\in {J}_{j},j\in {S}_{\mathbf{d}},\hfill \\ {c}_{j},\hfill & t\in {J}_{j},j\in {S}_{\mathbf{n}},\hfill \end{array}$

(39)

and the function

$\stackrel{\u02c6}{\gamma}(t)$ is given by

$\stackrel{\u02c6}{\gamma}(t)=\{\begin{array}{cc}{\varphi}_{j}(t)+{\sum}_{k\in {S}_{\mathbf{n}}}{a}_{k}{\gamma}_{j}^{[k]}(t),\hfill & t\in {J}_{j},j\in {S}_{\mathbf{d}},\hfill \\ {\tilde{\psi}}_{j}(t),\hfill & t\in {J}_{j},j\in {S}_{\mathbf{n}}.\hfill \end{array}$

(40)

Let

$c:=f(\mathrm{\infty})$ (unknown real constant) and

$g(z)$ be the analytic function defined on

*G* by

$g(z):=f(z)-c,\phantom{\rule{1em}{0ex}}z\in G.$

(41)

Then

$g(z)$ is analytic on

*G* with

$g(\mathrm{\infty})=0$. The function

$g(z)$ is a solution of the RH problem

$Re[A(t)g(\eta (t))]=\tilde{\gamma}(t)+h(t)+\sum _{j\in {S}_{\mathbf{d}}}{a}_{j}{\gamma}^{[j]}(t),\phantom{\rule{1em}{0ex}}t\in J,$

(42)

where the function

$A(t)$ is given by (10) and the function

$h(t)$ is defined by

$h(t)=\stackrel{\u02c6}{h}(t)-ccos\theta (t),\phantom{\rule{1em}{0ex}}t\in J.$

(43)