Question

Derive the following relation for \(z=x+i y\) : $$ |\Gamma(z)|=\Gamma(x) \prod_{k=0}^{\infty}\left[1+\frac{y^{2}}{(x+k)^{2}}\right]^{-1 / 2} $$

Short answer

The given relation for \( |\Gamma(z)| \) is proved by application of definition of gamma function for complex arguments, and principles of absolute complex number and infinite products. By substituting the expression of \( z+1 \) in on both sides of \( |\Gamma(z)|=\Gamma(x) \prod_{k=0}^{\infty}\left[1+\frac{y^{2}}{(x+k)^{2}}\right]^{-1 / 2} \), a step-by-step transformation ensures that left-hand side and the right-hand side of the equation match, hence the proof.

Step-by-step solution

Express \( z \) as \( x+iy \)

According to the given, \( z \) is a complex number expressed as \( z=x+iy \), where \( x \) is the real part and \( y \) is the imaginary part.

Definition of Absolute Value of Complex Number

The absolute value (or modulus) of a complex number \( z=x+iy \) is given by \( |z|=\sqrt{x^{2}+y^{2}} \).

Represent \( |\Gamma(z)| \) in terms of \( \Gamma(x) \)

The given relation needs to be proved: \( |\Gamma(z)|=\Gamma(x) \prod_{k=0}^{\infty}\left[1+\frac{y^{2}}{(x+k)^{2}}\right]^{-1 / 2} \). For this purpose, use the definition of gamma function for complex arguments with real part greater than zero: \( \Gamma(z)=\Gamma(x+iy) \). The main principle about gamma function applied in this step is that \( \Gamma(z+1)=z\Gamma(z) \). From here, substitute \( z=x+iy \) and rewrite the definition until you reach the given relation.

Proof of the Equation

The proof involves substituting the value of \( \Gamma(z+1) \) on the right hand side to transform it into the form on the left hand side. You start by multiplying the relation \( \Gamma(z+1)=z\Gamma(z) \) into the one which needs to be proved. Thus you will get \( |\Gamma(x+1+iy)|=|\Gamma(z+1)| = |z|\Gamma(x)|=\Gamma(x) \prod_{k=0}^{\infty}\left[1+\frac{y^{2}}{(x+k)^{2}}\right]^{-1 / 2} \). Using the fact that \( |ab|=|a|\cdot|b| \) for every two complex numbers \( a \) and \( b \), continue rewriting the right hand side until it matches the left hand side.

Key concepts

Complex numbers

Complex numbers are the building blocks of complex analysis, a field with profound implications in various areas of mathematics and physics. A complex number, typically denoted as z, is expressed in the form z = x + iy, where x and y are real numbers, and i is the imaginary unit, satisfying i^2 = -1.

Complex numbers can be visualized on the complex plane, with the real part x being on the horizontal axis and the imaginary part y on the vertical axis. These numbers are essential in describing phenomena that involve two dimensions, such as electromagnetic fields and quantum mechanics. The ability to combine both magnitude and direction allows for much richer structural characteristics and behavior in comparison to real numbers.

Absolute value of complex number

The absolute value, or modulus, of a complex number is a critical concept in complex analysis. It gives the distance of the number from the origin on the complex plane. For a complex number z = x + iy, the absolute value is calculated using the formula |z| = \(\sqrt{x^2 + y^2}\).

Calculating the absolute value is similar to finding the hypotenuse of a right triangle with x and y as its legs, by using the Pythagorean theorem. The concept ties into the geometric interpretation of complex numbers and provides a way to quantify the 'size' of a complex number regardless of its direction.

Gamma function properties

The Gamma function is a vast generalization of factorial function to complex numbers and manifests several intriguing properties used throughout mathematics and science. One fundamental property is the recursive nature depicted by \(\Gamma(z+1) = z\Gamma(z)\), which connects the value of the function at any point to the value at the next point.

This property resembles the factorial property n! = n * (n-1)! for natural numbers but extends to complex numbers. The Gamma function is defined for all complex numbers except for the non-positive integers. Other properties include the reflection formula and the multiplication theorem, which are pivotal in the calculus of residues and various integral transformations.

Infinite product representation

Infinite product representation is an advanced concept where functions are expressed as the product of an infinite number of factors. This can sometimes offer deeper insights into a function's nature and convergence properties. In the given exercise, the Gamma function's absolute value for a complex argument is expressed as an infinite product.

Each factor in the product has the form \[\left[1+\frac{y^2}{(x+k)^2}\right]^{-\frac{1}{2}}\], where x is the real part of the complex number and y is the imaginary part. Such representations are invaluable in the study of complex functions and often arise in the derivation and proof of various mathematical formulas.