Expressing sinh(z) in Complex Form: A Comprehensive Guide
Introduction
When working with complex numbers, expressing trigonometric functions in terms of exponential forms can be a valuable tool. The hyperbolic sine function, sinhz, is one such function that can be represented in various forms. In this article, we will explore how to express sinhz in its complex exponential form, which is a fundamental concept in complex analysis and has numerous applications in electrical engineering, physics, and mathematics.Understanding sinh(z)
The hyperbolic sine function, sinh(z), is defined as:sinhz (frac{e^z - e^{-z}}{2})
To express sinh(z) in its complex form, we first need to break down the terms in the exponential functions. Let z x iy, where x and y are real numbers and i is the imaginary unit. We can then write:sinhz (frac{e^z - e^{-z}}{2}) (frac{e^{x iy} - e^{-x-iy}}{2})
Expanding the exponentials, we have:sinhz (frac{e^x(cos(y) isin(y)) - e^{-x}(cos(-y) isin(-y))}{2})
Using the properties of the cosine and sine functions, we know that (cos(-y) cos(y)) and (sin(-y) -sin(y)). Therefore, the expression simplifies to:sinhz (frac{e^x(cos(y) isin(y)) - e^{-x}(cos(y) - isin(y))}{2})
Combining like terms, we get:sinhz (frac{e^xcos(y) i(e^xsin(y)) - e^{-x}cos(y) - i(e^{-x}sin(y))}{2})
Simplifying further, we can group the real and imaginary parts together:sinhz (frac{(e^xcos(y) - e^{-x}cos(y)) i(e^xsin(y) - e^{-x}sin(y))}{2})
Thus, the complex form of sinh(z) is:sinhz (sinh(x)cos(y) icosh(x)sin(y))
This form is particularly useful when dealing with complex arguments and allows us to separate the real and imaginary components of the function.Applications in Complex Analysis
The expression of sinh(z) in its complex form has several important applications in complex analysis. Here are a few examples: Electrical Engineering: In AC circuit analysis, the expression for sinh(z) is used to solve for currents and voltages in circuits with complex impedances. Physics: In quantum mechanics, the hyperbolic sine function appears in solutions to the Schr?dinger equation for certain potentials. Mathematics: The complex form of sinh(z) is essential in solving differential equations with complex coefficients and in studying conformal mappings in complex analysis.Beyond sinh(z): arcsinh(z)
In addition to expressing sinh(z) in its complex form, it is also useful to understand the inverse function, arcsinh(z). The inverse hyperbolic sine function can be expressed as:arcsinh(z) sign(x) log |z (sqrt{|z|^2 - 1})| i(frac{pi}{2}) (2n 1) for z ≠ 0
where n is an integer, and sign(x) is the sign of the real part of z. This expression is derived from the logarithmic form of the inverse hyperbolic sine function and is particularly useful in evaluating the inverse of sinh(z) for complex arguments.