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)