The complex hyperbolic functions are defined as coshz=(e^z+e^-z)/(2), sinhz=(e^z-e^-z)/(2) (a) W

step 1 We need to show that these two pieces are true. Let's start with that kosh of z1 is equal to cos

The complex hyperbolic functions are defined as...

The complex hyperbolic functions are defined as $\cosh z=\frac{e^z+e^{-z}}{2}, \sinh z=\frac{e^z-e^{-z}}{2}$ (a) Write out the formulas for their real and imaginary parts in terms of $z=x+\mathrm{i} y$. (b) Prove that $\cos \mathrm{i} z=\cosh z$ and $\sin \mathrm{i} z=\mathrm{i} \sinh z$.

The complex hyperbolic functions are defined as $\cosh z=\frac{e^z+e^{-z}}{2}, \sinh z=\frac{e^z-e^{-z}}{2}$
(a) Write out the formulas for their real and imaginary parts in terms of $z=x+\mathrm{i} y$.
(b) Prove that $\cos \mathrm{i} z=\cosh z$ and $\sin \mathrm{i} z=\mathrm{i} \sinh z$.

Key Concepts

Hyperbolic-Trigonometric Identities

This key concept highlights the relationships between hyperbolic and trigonometric functions, especially when the argument is multiplied by the imaginary unit. Identities like cos(i z) = cosh z and sin(i z) = i sinh z emerge from substituting complex arguments into the definitions of cosine and sine and then employing Euler's formula. These identities exemplify the deep connections between the hyperbolic and trigonometric function families.

Euler's Formula

Euler's formula, e^(iy) = cos y + i sin y, is a cornerstone in complex analysis that connects complex exponentials with trigonometric functions. It is essential for converting and simplifying expressions involving complex exponentials, thereby facilitating the extraction of real and imaginary parts in complex functions such as those for hyperbolic sine and cosine.

Complex Hyperbolic Functions

This concept refers to the functions defined using exponentials that generalize the hyperbolic cosine and sine to complex arguments. These functions are defined by cosh z = (e^z + e^(–z))/2 and sinh z = (e^z – e^(–z))/2, and their properties extend naturally into the complex plane, allowing analysis of both their magnitude and periodic behavior in a complex setting.

Decomposition into Real and Imaginary Parts

When a complex variable is expressed as z = x + iy, functions of z can be separated into their real and imaginary parts. This process involves rewriting the exponential functions using identities from complex analysis to identify components that correspond to the cosine and sine functions. This decomposition is crucial for understanding the behavior of complex functions and for proving relationships between various complex functions.

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