4.2 Trigonometric and hyperbolic functions

Complex trigonometric and hyperbolic functions extend real-valued functions to the complex plane. They're defined using the complex exponential function and Euler's formula, maintaining key properties like periodicity and identities.

These functions play a crucial role in complex analysis, allowing us to solve equations and model phenomena in the complex domain. Understanding their definitions, properties, and relationships is essential for working with complex-valued functions.

Trigonometric functions in the complex plane

Complex exponential function and Euler's formula

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• The complex exponential function $e^{z}$ is defined as $e^{z} = e^{x+iy} = e^x(\cos y + i \sin y)$ for any complex number $z = x + iy$
• Euler's formula, $e^{i\theta} = \cos \theta + i \sin \theta$, relates the complex exponential function to the complex trigonometric functions
• Example: $e^{i\pi/4} = \cos(\pi/4) + i\sin(\pi/4) = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}$

Definitions of complex trigonometric functions

• The complex sine function is defined as $\sin z = \frac{e^{iz}-e^{-iz}}{2i}$ for any complex number $z$
• The complex cosine function is defined as $\cos z = \frac{e^{iz}+e^{-iz}}{2}$ for any complex number $z$
• The complex tangent function is defined as $\tan z = \frac{\sin z}{\cos z} = \frac{e^{iz}-e^{-iz}}{i(e^{iz}+e^{-iz})}$ for any complex number $z$ where $\cos z \neq 0$
• The complex cotangent, secant, and cosecant functions are defined as $\cot z = \frac{\cos z}{\sin z}$, $\sec z = \frac{1}{\cos z}$, and $\csc z = \frac{1}{\sin z}$, respectively, for any complex number $z$ where the denominator is non-zero
• Example: $\sin(1+2i) = \frac{e^{i(1+2i)}-e^{-i(1+2i)}}{2i} \approx 3.1657 + 1.9596i$

Periodicity of complex trigonometric functions

• The complex trigonometric functions are periodic with period $2\pi$
• $\sin(z+2\pi) = \sin z$, $\cos(z+2\pi) = \cos z$, and $\tan(z+\pi) = \tan z$ for any complex number $z$
• This periodicity allows for the extension of trigonometric functions to the complex plane while maintaining their fundamental properties
• Example: $\sin(z) = \sin(z+2\pi) = \sin(z+4\pi) = \ldots$

Complex hyperbolic functions

Definitions of complex hyperbolic functions

• The complex hyperbolic sine function is defined as $\sinh z = \frac{e^{z}-e^{-z}}{2}$ for any complex number $z$
• The complex hyperbolic cosine function is defined as $\cosh z = \frac{e^{z}+e^{-z}}{2}$ for any complex number $z$
• The complex hyperbolic tangent function is defined as $\tanh z = \frac{\sinh z}{\cosh z} = \frac{e^{z}-e^{-z}}{e^{z}+e^{-z}}$ for any complex number $z$ where $\cosh z \neq 0$
• The complex hyperbolic cotangent, secant, and cosecant functions are defined as $\coth z = \frac{\cosh z}{\sinh z}$, $\operatorname{sech} z = \frac{1}{\cosh z}$, and $\operatorname{csch} z = \frac{1}{\sinh z}$, respectively, for any complex number $z$ where the denominator is non-zero
• Example: $\sinh(1+i) = \frac{e^{1+i}-e^{-(1+i)}}{2} \approx 0.6349 + 1.2985i$

Expressing complex hyperbolic functions using the complex exponential function

• The complex hyperbolic functions can be expressed in terms of the complex exponential function
• $\sinh(x+iy) = \sinh x \cos y + i \cosh x \sin y$
• $\cosh(x+iy) = \cosh x \cos y + i \sinh x \sin y$
• These expressions allow for the evaluation of complex hyperbolic functions using their real and imaginary parts
• Example: $\cosh(2+3i) = \cosh 2 \cos 3 + i \sinh 2 \sin 3 \approx -3.7245 + 0.5118i$

Identities for complex functions

Addition formulas for complex trigonometric and hyperbolic functions

• The addition formulas for complex trigonometric functions are:
  • $\sin(z_1+z_2) = \sin z_1 \cos z_2 + \cos z_1 \sin z_2$
  • $\cos(z_1+z_2) = \cos z_1 \cos z_2 - \sin z_1 \sin z_2$
  • $\tan(z_1+z_2) = \frac{\tan z_1 + \tan z_2}{1 - \tan z_1 \tan z_2}$ for any complex numbers $z_1$ and $z_2$ where the denominators are non-zero
• The addition formulas for complex hyperbolic functions are:
  • $\sinh(z_1+z_2) = \sinh z_1 \cosh z_2 + \cosh z_1 \sinh z_2$
  • $\cosh(z_1+z_2) = \cosh z_1 \cosh z_2 + \sinh z_1 \sinh z_2$
  • $\tanh(z_1+z_2) = \frac{\tanh z_1 + \tanh z_2}{1 + \tanh z_1 \tanh z_2}$ for any complex numbers $z_1$ and $z_2$ where the denominators are non-zero
• Example: $\sin(1+i) + \sin(2-i) = \sin(1+i)\cos(2-i) + \cos(1+i)\sin(2-i) \approx 1.4031 + 1.3191i$

Double angle formulas for complex trigonometric and hyperbolic functions

• The double angle formulas for complex trigonometric functions are:
  • $\sin 2z = 2 \sin z \cos z$
  • $\cos 2z = \cos^2 z - \sin^2 z$
  • $\tan 2z = \frac{2 \tan z}{1 - \tan^2 z}$ for any complex number $z$ where the denominators are non-zero
• The double angle formulas for complex hyperbolic functions are:
  • $\sinh 2z = 2 \sinh z \cosh z$
  • $\cosh 2z = \cosh^2 z + \sinh^2 z$
  • $\tanh 2z = \frac{2 \tanh z}{1 + \tanh^2 z}$ for any complex number $z$ where the denominators are non-zero
• Example: $\cos(2(1+i)) = \cos^2(1+i) - \sin^2(1+i) \approx 0.8370 - 0.9888i$

Pythagorean identities for complex trigonometric and hyperbolic functions

• The Pythagorean identities for complex trigonometric functions are:
  • $\sin^2 z + \cos^2 z = 1$
  • $1 + \tan^2 z = \sec^2 z$ for any complex number $z$ where the functions are defined
• The Pythagorean identities for complex hyperbolic functions are:
  • $\cosh^2 z - \sinh^2 z = 1$
  • $\tanh^2 z + \operatorname{sech}^2 z = 1$ for any complex number $z$ where the functions are defined
• These identities extend the fundamental relationships between trigonometric and hyperbolic functions to the complex plane
• Example: $\sin^2(2+3i) + \cos^2(2+3i) = 1$

Solving equations with complex functions

Solving equations with a single complex trigonometric or hyperbolic function

• To solve equations involving a single complex trigonometric or hyperbolic function, use the inverse function to solve for the variable
• If $\sin z = w$, then $z = \arcsin w + 2\pi n$ for any integer $n$
• Similarly, use the inverse functions $\arccos$, $\arctan$, $\operatorname{arccot}$, $\operatorname{arcsec}$, $\operatorname{arccsc}$, $\operatorname{arcsinh}$, $\operatorname{arccosh}$, $\operatorname{arctanh}$, $\operatorname{arccoth}$, $\operatorname{arcsech}$, and $\operatorname{arccsch}$ to solve equations involving the corresponding complex trigonometric or hyperbolic functions
• Example: If $\cosh z = 2$, then $z = \operatorname{arccosh} 2 \approx 1.3170$

Solving equations with multiple complex trigonometric or hyperbolic functions

• For equations involving multiple complex trigonometric or hyperbolic functions, use the identities to express the equation in terms of a single function, then solve for the variable using the inverse function
• Be aware of the domain and range of the functions involved and consider any restrictions on the variable
• Some equations may have multiple solutions or no solutions depending on the values of the constants and the functions involved
• Example: If $\sin z + \cos z = 1$, then $\sin z = 1 - \cos z$, and substituting this into the Pythagorean identity gives $\cos z = \frac{\sqrt{2}}{2}$. Thus, $z = \arccos(\frac{\sqrt{2}}{2}) + 2\pi n \approx \frac{\pi}{4} + 2\pi n$ for any integer $n$

Graphical methods for solving complex trigonometric and hyperbolic equations

• Graphical methods, such as plotting the functions on the complex plane, can be used to visualize and approximate the solutions to complex trigonometric and hyperbolic equations
• By plotting both sides of the equation and observing the intersection points, one can estimate the solutions to the equation
• Graphical methods can be particularly helpful when dealing with equations that are difficult to solve analytically or have multiple solutions
• Example: To solve $\sin z = z$, plot both $\sin z$ and $z$ on the complex plane and find the intersection points. The solutions will be the complex numbers corresponding to these intersection points