To determine if $u = \cos x \cosh y$ is harmonic, we must verify that it satisfies Laplace's equation, which in two dimensions is given by:

$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$

1. Compute the partial derivatives of $u$:

   Given: $u = \cos x \cosh y$

   Let's find $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$:

   • First derivative with respect to $x$: $\frac{\partial u}{\partial x} = -\sin x \cosh y$

   • Second derivative with respect to $x$: $\frac{\partial^2 u}{\partial x^2} = -\cos x \cosh y$

   • First derivative with respect to $y$: $\frac{\partial u}{\partial y} = \cos x \sinh y$

   • Second derivative with respect to $y$: $\frac{\partial^2 u}{\partial y^2} = \cos x \cosh y$

2. Verify Laplace's equation:

   Now, we add $\frac{\partial^2 u}{\partial x^2}$ and $\frac{\partial^2 u}{\partial y^2}$: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = -\cos x \cosh y + \cos x \cosh y = 0$

   Since this sum is zero, $u = \cos x \cosh y$ is indeed a harmonic function.

3. Find the harmonic conjugate $v(x, y)$:

   To find the harmonic conjugate $v$, we need to satisfy the Cauchy-Riemann equations: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

   From our calculations: $\frac{\partial u}{\partial x} = -\sin x \cosh y$ $\frac{\partial u}{\partial y} = \cos x \sinh y$

   Let $\frac{\partial v}{\partial y} = -\sin x \cosh y$. Integrating with respect to $y$ gives: $v(x, y) = -\sin x \sinh y + C(x)$

   To determine $C(x)$, we use $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$: $\frac{\partial v}{\partial x} = -\cos x \sinh y$

   This matches, so we conclude: $v(x, y) = -\sin x \sinh y$

4. Form the analytic function $f(z) = u + iv$:

   The analytic function $f(z)$ is: $f(z) = u + iv = \cos x \cosh y - i \sin x \sinh y$

   Using the identity $z = x + iy$, we can rewrite this as: $f(z) = \cos z$

Thus, $f(z) = \cos z$ is the analytic function corresponding to the harmonic function $u = \cos x \cosh y$.

Would you like further details on any step?

Here are five related questions for deeper exploration:

1. What other functions satisfy Laplace's equation and are therefore harmonic?
2. Can we use the Cauchy-Riemann equations to find harmonic conjugates of any harmonic function?
3. How would we represent $f(z) = \cos z$ in terms of real and imaginary components?
4. How can we confirm the analyticity of $f(z) = \cos z$ in complex analysis?
5. How does the harmonic conjugate relate to contour integration?

Tip: Always remember that a harmonic function can generate an analytic function if it has a harmonic conjugate that satisfies the Cauchy-Riemann equations.