The formulas cosz=(e^i z+e^-i z)/(2) and sinz=(e^i z-e^-i z)/(2 i) serve to define the basic com

step 1 In this problem, we have the complex number as e power of iz. So, taking z as x plus iy, this ca

The formulas cosz=(e^i z+e^-i z)/(2) and sinz=(e^i z-e^-i...

The formulas $\cos z=\frac{e^{\mathrm{i} z}+e^{-\mathrm{i} z}}{2}$ and $\sin z=\frac{e^{\mathrm{i} z}-e^{-\mathrm{i} z}}{2 \mathrm{i}}$ serve to define the basic complex trigonometric functions. Write out the formulas for their real and imaginary parts in terms of $z=x+\mathrm{i} y$, and show that $\cos z$ and $\sin z$ reduce to their usual real forms when $z=x$ is real. What do they become when $z=\mathrm{i} y$ is purely imaginary?

Key Concepts

Reduction to Special Cases

In complex analysis, examining how complex functions behave for special cases, such as when the argument is purely real (y = 0) or purely imaginary (x = 0), is a vital technique. When z = x (a real number), the definitions of cos?z and sin?z reduce to the classical real trigonometric functions. In contrast, when z = i?y, these functions transform into expressions involving hyperbolic functions, specifically cos(i?y) = cosh?y and sin(i?y) = i?sinh?y. This differentiation emphasizes the dual nature of these functions in complex analysis.

Hyperbolic Functions

Hyperbolic functions, such as cosh?y and sinh?y, arise naturally when separating the exponential representations of trigonometric functions for complex arguments. When the imaginary unit is factored into the expressions, the definitions of cosh?y and sinh?y help rewrite the complex sine and cosine in a form that clearly shows the contribution of the imaginary part. These functions share many similarities with their trigonometric counterparts and are widely used in both theoretical and applied mathematics.

Separation of Real and Imaginary Parts

When dealing with functions of a complex variable, it is often useful to express the function in terms of its real and imaginary components. For a complex number expressed in the form z = x + i?y, writing the functions in terms of x and y helps in understanding how the function operates differently on the real and imaginary parts. This process is essential for analyzing limits, continuity, and differentiability in complex analysis.

Euler's Formula

Euler's formula, which states that e^(i?) = cos ? + i sin ? for real ?, is a cornerstone of complex analysis. It establishes a deep connection between exponential functions and trigonometric functions, allowing a seamless transition from real to complex analysis. This formula underpins many definitions and proofs within the subject, including the representation of complex trigonometric functions.

Complex Exponential Function

The complex exponential function, defined by e^(z) for complex z, is a fundamental extension of the real exponential function into the complex plane. It encapsulates both exponential growth (or decay) and rotation in the complex plane, and its properties are pivotal in many areas of mathematics including differential equations, Fourier analysis, and complex dynamics.

Complex Trigonometric Functions

Complex trigonometric functions are defined via their representations in terms of complex exponentials. For example, the functions cos?z and sin?z are defined respectively as (e^(iz) + e^(?iz))/2 and (e^(iz) ? e^(?iz))/(2i). This definition naturally extends these functions from real to complex arguments, preserving many of their algebraic properties while introducing new behaviors in the complex plane.

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